Sketch the region Dbetween y= x2 and y= x(1 x). 14 a Evaluate: C R PS HE i 42 ii (-4)2 M AT I C A 2 b If a = 16, write down the possible values of a. Find an equation for the circle that has center s21, 4d and passes through the point s3, 22d. 6 #22 for correctness and others for completion. com/multiple-integrals-course Learn how to use double integrals to find the surface area. 2x 3y 6 0 29. For more videos like this one, please. x2 + Y2 over the region 1 x2 + e. Find the area of the surface. (a) the part of the plane 2x+5y +z = 10, that lies inside the cylinder x2 +y2 = 9. ∂z = 10x+14y +6z. (a)Write down a parametrization of S. variables, we get: z=6-2y-3x. ±4 ±2 0 2 4 x ±4 ±2 0 2 4 y ±4. m Find the ux of F = _. Find the midpoint of the chord intercepted by the line 5x -y + 9=0 on the circle x2 + y2 = 18. Note that the surface S consists of a portion of the paraboloid z = x2 +y2 and a portion of the plane z = 4. (iii) The second plane must have the same normal vector hence the same coefﬁcients for x;y;z. Hold it! Yes, it is a triangle, and therefore half the area of the rectangle, but that's not what you're supposed to be learning from this question. if you give your honest and detailed thoughts then people will find new books that are. Find the ﬂux of F = xi +yj +zk through S; take the positive side of S as the one where the normal points "up". Find the surface area of the part of the plane 5x+2y+z=4 that lies inside the cylinder x^2+y^2=9. Find the surface area of the part of the plane 5x + 5y + z = 25 that lies above the triangle formed by the three points (4,2,0)(8,2,0)(8,6,0) - 7169673. Solution: This is Problem 15. Geometrically, the line integral is the area of a cylinder of. Find the surface area of the surface with parametric equations and S is the part of the paraboloid z =9−x2 −y2 6. The part of the sphere x y z z2 2 2 4 that lies inside the paraboloid z x y22. Find the area of the surface. First we parametrize the whole cone by r(u; ) = ucos i+ usin j+ uk and again we look for the restrictions corresponding to the part also within the sphere. Given that the integral the value of 19. Double integral Riemann sum. b Write the length of each room in simplest surd form. Find the area of the part of the surface z = 1 + 3x + 2y^2 that lies above the triangle? in the x-y plane with vertices (0; 0), (0; 1) and (2; 1). Evaluate the triple integral. Homework Solutions MATH 32B-2 (18W) Problem 10 (16. Sketch the function. Answer: The shadow of the plane will just be the disk x 2+y ≤ 9. Let's take a look at a couple of examples. Find the area of the surface. We are told that 78 = 2w + 6. plane 3x+2y+z=6. DEPARTMENT OF MATHEMATICS, IIT MADRAS MA 1020: Calculus-I: Problem Sheet - 4 1. Find the area of the surface z= 1 + x2 above the region Rin the xy-plane bounded by y= x; y= 0 and x= 1. 5 Problem 2E. Question: Find the surface area of that part of the plane 5x + 2y + z = 10 that lies inside the elliptic cylinder (x 2)/25 + (y 2)/64 = 1. a plane -1(x + 3) - 10(y - 3) + 6(z - 2) = 0 -x - 10y + 6z = -15 or x + 10y - 6z = 15 Surface Area To find the surface area of a parametrically defined surface, we proceed in a similar way as in the case as a surface defined by a function. y2 dx+ 3xydy where Cis the positively oriented boundary of the semi-annular region Rin the upper half-plane between the circles x 2+ y = 1 and x2 + y = 4. (1 pt) Use Stoke’s Theorem to evaluate Integral_C (F · dr) where F(x,y, z) = x i+y j+3(x^2 +y^2) k and C is the boundary of the part of the paraboloid where z = 9?x^2 ?y^2 which lies above the xy-plane and C is oriented counterclockwise when viewed from. Evaluate the triple integral. Therefore, it is clear that the region S is the ﬁrst octant of an ellipsoid. I create online courses to help you rock your math class. Then we are forced to have z= rcos( ) + 3. Use the product rule to compute the derivative of f (x) = (2x - 3)2. Textbook solution for Calculus: Early Transcendentals 8th Edition James Stewart Chapter 15. The volume is computed over the region D defined by 0≤x≤2 and 0≤y≤1. Calculate the attitude ? between the given airplane 4x + 5y + z = 4 and the xy-airplane. e Find 3 −27. (a) Find a parameterization for the hyperboloid x2 + y2 z2 = 25. The part of the sphere x² + y² + z² = a² that lies within the cylinder x² + y² = ax and above the xy-plane a²(π - 2). Multivariable Calculus: Find the area of the surface z = (x^2 + y^2)^1/2 over the unit disk in the xy-plane. Now the plane intersects this cylinder at an angle. y x z FIGURE 12 19. Test closed curve. Find the surface area of the part of the plane 2x+ 5y+ z= 10 that lies inside the cylinder x2 + y2 = 9:(HW 15) 25. Instead of calculating line integral. (a) RRR E Find the volume of the region inside the cylinder x 2+ y = 9, lying above the xy-plane, and below the plane z = y +3. Answered by Penny Nom. Find the ﬂux of → F = x,y2,z upward through the ﬁrst-octant part S of the cylindrical surface x2 +z2 = a2 for 0 6 y 6 b. 43 dened parametrically by r=ucosvi+usinvj+u2k, SECTION 15. Fund Raiser Lesson 2. Find the center and radius of the circle with equation x. Math 232 Practice Exam #3 Solutions 1. (a) the part of the plane 2x+5y +z = 10, that lies inside the cylinder x2 +y2 = 9. 0 m, and between the sheets for the following situations. Find the surface area of the part of the paraboloid z = 9 - x2 - y2 that lies above the plane z = 3. Joined Jan 28, 2006 Messages 134. ∫ C F ⋅ d s. Math 2263 Quiz 10 26 April, 2012 Name: 1. You're having trouble because you're trying to describe the surface in rectangular coordinates, when instead there is an obvious parametrization using polar coordinates. Sketch the surface. Get real results without ever leaving the house. 7 Lagrange Multipliers. Find the mass of that part of the surface z= xythat lies within one unit of the z-axis if the density at the point (x;y) is given by (x;y) = x2 + y2. •The elliptical region S in the plane y + z = 2 that is bounded by C. Find the volume of the region that lies above the paraboloid z= 2x2 +2y2 and lies below the cone z= 2 p x2 + y2. Math 32B Discussion Session Week 10 Notes March 14 and March 16, 2017 Find the area of the region in the rst quadrant enclosed by the ellipses x2 4 +y 2 = 1 and x2 7. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these. 44-10, the base is the circle x2 + y2 = I in the ry-plane, the top is the plane x + z = 1. Math 114 Practice Problems for Test 2 Comments: 1. It x—3 for x 23. The surface bounding the solid from above is the graph of a positive function z= f(y) z)-plane (which is equal to the area of the front face of the solid), which is then given by the integral of fover the interval [0;a]. The quantity G(yi)∆y can be interpreted as the volume of a solid with face area G(yi) and thickness ∆y. First we parametrize the whole cone by r(u; ) = ucos i+ usin j+ uk and again we look for the restrictions corresponding to the part also within the sphere. 4 Surface Integrals 573 44 Show that the spin field S does work around every simple inside R can be squeezed to a point without leaving R. As seen in Fig. in segment form. ) is written as y = 2 - 2x. The base of the cylinder is shown in the sketch below. Remember that the first octant is the portion of the xyz-axis system in which all three variables are positive. To normalize the answer, make …. You may use that the surface area of a cone of height hand base radius ris given by ˇr(r+ p h2 + r2). The part of the surface z = 10 that is above the square −1 ≤ x ≤ 1, −2 ≤ y ≤ 2. 825cm Answer: The area of the equilateral triangle is 10. Set up a triple integral in spherical coordinates that gives the volume of the solid that lies outside the cone z = 3x2 +3y2 and inside the hemisphere z =. (6 Points) Find the center and radius of the following sphere x2 +y2 +z2 6x+4z 3 = 0. SOLUTION KEYS FOR MATH 105 HW (SPRING 2013) STEVEN J. To show that the flux across \(S\) is the charge inside the surface divided by constant \(\epsilon_0\), we need two intermediate steps. The cylinder x2 +y2 = x 1. Plane: 10x+3y+z=10 Cylinder: (x^2)/81+(y^2)/100 = 1. A)1225π B) 245 2 π C)175π D) 35 2 π 27). I Review: Double integral of a scalar function. Thread starter mathstresser; Start date Oct 23, 2006; M. Math 2263 Quiz 10 26 April, 2012 Name: 1. We can write z = 10 2x 5y = f (x;y). 1VectorFieldspage554CHAPTER15VECTORCALCULUS15. Answered by Penny Nom. Answer to Find the surface area of the part of the plane 4x+1y+z=1 that lies inside the cylinder x^2+y^2=9. The part of the plane 5x + 3y - z + 6 = 0 that lies above the rectangle [1, 4] × [2, 6]. Example The equation z = 3 describes a plane that is parallel to the xy-plane, and is 3 units \above" it; that is, it lies 3 units along the positive z-axis from the xy-plane. ranges here in the interval 0 \le x \le 1, and the variable y. The cross sections of the solid perpendicular to the y-axis are squares. Example 2 (a) Find parametric equations for the line through (5,1,0) that is perpendicular to the plane 2x − y + z = 1 A normal vector to the plane is:. •The projection D of S on the xy-plane is the disk x2 + y2 ≤ 1. Selected Solutions, Sections 16. y 3 x 1 4 27. Let f(x;y) = xy. curlF dS where z) and S is the part of the sphere + + z2 = 36 that lies inside the cylinder + y 2 — 9 and above the xy-plane. Use a double integral to calculate the volume of the solid under the plane z =2x + y and over the rectangle R = {(x,y):3 x 5,1 y 2}. Solution: Step 1: Given that the radius is 2cm. Thus this is the surface area of the part of the surface z= 6 3x 2yover. (a) The part of the cone z = p x2 + y2 below the plane z = 3. Evaluate the surface area of the part of the surface z= p x2 + y2 between the planes z= 1 and z= 2. Find the area of the surface. Answer: (x−1)2+(y −1)2+(z −1)2 = x2 +y2+z2 gives −2x+1−2y +1−2z +1 = 0 or 2x+2y +2z = 3. View ps1 from BUSINESS 1ba3 at McMaster University. radical 139. 3x 2y 5 0 16. x + 2y + 3z = 1. Three collinear points: 2014-08-28: From jhanavi: p,q,r are three collinear points. The surface is the graph of the function f(x,y)=cos2x+sin2y. Thus A = R 2π 0 R 3 0 √ 1+4+25rdrdθ = 9 √ 30π. Find the area of the surface: Solution: Let z = f(x,y), (x,y) ∈ D. The part of the plane. The part of the hyperbolic paraboloid. Solutions for practice problems, Fall 2016 Qinfeng Li December 5, 2016 Problem 1. The part of the hyperbolic paraboloid z = y2 − x2 that lies between the cylinders x2 + y2 = 1 and x2 + y2 = 9. Find the surface area of the surface with parametric equations x = uv y = u+v z = u−v (x,y,z)=yzi+xzj+xyk and S is the part of the paraboloid z =9−x2 −y2 6. As seen in Fig. DEPARTMENT OF MATHEMATICS, IIT MADRAS MA 1020: Calculus-I: Problem Sheet - 4 1. We can write z = 10 2x 5y = f (x;y). Refer to section 4. Suppose that the temperature on this sphere is given. Go back to your professor (or your book) and find how to integrate over irregular areas. Answer: The picture looks like this. Use the product rule to compute the derivative of f (x) = (2x - 3)2. The complete proof of Stokes’ theorem is beyond the scope of this text. Surface worksheet solutions Parameterize the part of the plane z = x+ 3 that lies inside the cylinder x 2+ y = 9. The cross sections of the solid perpendicular to the y-axis are squares. 8 a)Find the distances of P = (12,5,0) to each of the 3. When two sphere intersect they form a circle if you join all the points of intersection, that is the common circle. (a) Find the volume of the region inside the cylinder x 2+ y = 9, lying above the xy-plane, and below the plane z = y +3. ranges in the interval 0 \le y \le 2 – 2x. Find the volume of the solid bounded by the paraboloids z= 2x2 + y2 and z= 27 x2 2y2. Use a double integral to calculate the volume of the solid under the plane z =2x + y and over the rectangle R = {(x,y):3 x 5,1 y 2}. In cylindrical coordinates, the volume of a solid is defined by the formula. 0 m, and between the sheets for the following situations. (a)Write down a parametrization of S. Let Dbe the solid that lies inside the cylinder x2+y2 = 1, below the cone z= p 4(x2 + y2) and above the plane z= 0. This cylinder can be parameterized by R~( ;z) = h3cos ;3sin ;zi for 0 2ˇand 0 z 5. Calculate the surface area of the surface obtained by revolving the curve y= x3 3 around the x-axis for 1 x 2. (b) that part of the elliptical paraboloid x + y 2+ 2z = 4 that lies in front of the plane x = 0. Solving this equation, we see that the width of the tennis court is (78 – 6) ÷ 2 = 72 ÷ 2 = 36 ft. The part of the plane z=6+5x+2y that lies above the rectangle [0,9] x [1,4]. Solution: Surface lies above the disk x 2+ z in the. Requires a little geometric insight. We can write the above integral as an iterated double integral. 190 Chapter 9 Applications of Integration It is clear from the ﬁgure that the area we want is the area under f minus the area under g, which is to say Z2 1 f(x)dx− Z2 1 g(x)dx = Z2 1 f(x)−g(x)dx. Unformatted text preview: that lies above the xy-plane. Find Z Z R xexydxdy whereR= [0;2] [0;1]. y = x2 /y(1 + x3 ) 4. This is the surface integral of the function x 2+ y over the part of the surface. HW11: I graded 16. Can someone help me with this/how to do one like it. find coordinates of R Answered by Penny Nom. The annular region Dcan be described in polar coordinates by. Completing the squares: 0 = x2 + y2 + z2 6x+ 4z 3 = (x2 6x+ 9) + y2 + (z2 + 4z+ 4) 3 9 4 = (x 3)2 + y2 + (z+ 2)2 16: So, the equation of the sphere is (x 22) 2+y +(z ( 2)) = 42, the center is (3;0; 2) and radius 4. Let Sbe the part of the sphere x2 +y2 +z2 = 25 that lies below the plane z= 4, oriented so that the unit normal vector at (0;0; 5) is h0;0; 1i. The same thing happens on the other two squares so we have that the whole ux integral is zero. After computing, we re-derive the area formula. Example 2 (a) Find parametric equations for the line through (5,1,0) that is perpendicular to the plane 2x − y + z = 1 A normal vector to the plane is:. Find a parametric representation for the surface consisting of that part of the elliptic paraboloid x +y2 +2z2 = 4 that lies in front of the plane x = 0. Studyres contains millions of educational documents, questions and answers, notes about the course, tutoring questions, cards and course recommendations that will help you learn and learn. 10 Describe the set xy = 0. (b) Find an expression for a unit normal to this surface. The surface bounding the solid from above is the graph of a positive function z= f(y) z)-plane (which is equal to the area of the front face of the solid), which is then given by the integral of fover the interval [0;a]. You may use that the surface area of a cone of height hand base radius ris given by ˇr(r+ p h2 + r2). Find the volume of the solid inside the cylinder x 2+ y = 4 and between the cone z= 5 p x2 + y2 and the xy-plane. 1− x2 −z4/4, we have the upper surface y2 9 + x2 + z2 4 = 1, which is an ellipsoid. Find the value of correct to four decimal places, where is the part of the paraboloid that lies above the -plane. Axis of the cylinder is the z axis (x=0,y=0) and the cylinder is a circle of radius 5 (sqrt(25)) about this axis. Problem 3 Find the surface area of the part of z = 1−x2 −y2 that lies above the xy-plane. Studyres contains millions of educational documents, questions and answers, notes about the course, tutoring questions, cards and course recommendations that will help you learn and learn. I got 16/3 from using triple integrals, and from using a visual approach. Above xy plane, which is z=0 And bounded by cylinder x^2+y^2=2x. --Lambiam Talk 22:20, 4 May 2006 (UTC). The annular region Dcan be described in polar coordinates by. The part of the plane 5x + 3y - z + 6 = 0 that lies above the rectangle [1, 4] × [2, 6]. The Curved Surface Area of a Surface of Revolution. variables, which can be obtained by using the constraint that (x;y;z) lies on the given plane. Find the area of the surface. Find parametric equations for the surface obtained by rotating the curve y = e-x, 0 ~ x ~ 3, about the x-axis and use them to graph the surface. Math 114 Practice Problems for Test 2 Comments: 1. Call this region S. Live Music Archive. Example 2 (a) Find parametric equations for the line through (5,1,0) that is perpendicular to the plane 2x − y + z = 1 A normal vector to the plane is:. The magnitude of the generalised force is 2(1-4) O for xo then 20. Now the plane intersects this cylinder at an angle. Find the area of the surface. Solution: Let S1 be the part of the paraboloid z = x2 + y2 that lies below the plane z = 4, and let S2 be the disk x2 +y2 ≤ 4, z = 4. Volume in the rst octant bounded by cylinder z = 16 x2 and the plane y = 5. Thus, we can plug in 3x 22y for z 1 into d and so the point we are looking for is where f(x;y) = (x+ 1) 2+ (y 2) + ( 3x 2y)2 = 10x2 + 12xy + 5y2 + 2x 4y + 5 attains a global minimum. Computethedoubleintegral Z 1 0 Z ex x xy2 dydx Problem 14. {image} {image} {image} {image} {image} 3. where E is the solid that lies within the cylinder x 2+ y2 = 1, above the plane z = 0, and below the cone z2 = 4x +4y2. is part of the answer. x y z z = 9− 2x r = 2cosθ Solution: ˚ D dV = ˆπ/2 −π/2 ˆ 2cosθ 0 ˆ 9−2x 0 rdzdrdθ Of course, 9 − 2x = 9− 2rcosθ. •The elliptical region S in the plane y + z = 2 that is bounded by C. The total flux through the surface is This is a surface integral. Answer: The picture looks like this. 8 a)Find the distances of P = (12,5,0) to each of the 3. The two parts of this product have useful meaning: (b− a)(d− c) is of course the area of the rectangle, and the double sum adds up mn terms of the form f(xj,yi)∆x∆y, which is the height of the surface at a point times the area of one of the small rectangles into which we have divided the large rectangle. Solutions for practice problems, Fall 2016 Qinfeng Li December 5, 2016 Problem 1. We write the equation of the plane ABC. Find the area of the part of the surface z= x2 + y2 between the planes z= 1 and z= 2. Evaluate the surface area of the part of the surface z= p x2 + y2 between the planes z= 1 and z= 2. The set is a union of two planes. The equation of the tangent plane is This last equation is the same as the equation for the plane with a replaced by the x derivative and b replaced by the y derivative. {image} {image} {image} {image} {image} 3. 18 Find a parametric representation for the surface which is the lower half of the ellipsoid 2x2 + 4y2 + z2 = 1 The lower half of the ellipsoid is given by z= p 1 2x2 4y2:. Math 2263 Quiz 10 26 April, 2012 Name: 1. See attached file for full problem description. ±4 ±2 0 2 4 x ±4 ±2 0 2 4 y ±4. ) could be evaluated directly, however, it’s easier to use Stokes’ Theorem. Setup a double integral that represents the surface area of the part of the paraboloid Z: = 16 A?— 3x2 A?— 3y2 that lies above the xy-plane. Use the chain rule to find δz δr and δz δθ. Z 2ˇ 0 Z ˇ 0 csin˚d˚d = 4ˇc; which doesn’t depend on R. The portion of the plane 2x − 2y + z = 1 lying in the ﬁrst octant forms a triangle S. the given surface is the cylinder that consists of all the vertical lines passing through the points of the circle x2 +y2 = x in the xy-plane (see Figure 1. 14 a Evaluate: C R PS HE i 42 ii (-4)2 M AT I C A 2 b If a = 16, write down the possible values of a. Consider the sphere x 2+ y + z2 = 9 which models an imaginary planet. (a) Let a,b > 0 and let a point of mass m move along the curve in R3 deﬁned by c(t)=(acost,bsint,t2). I Review: Double integral of a scalar function. Find the area of th&surface x2 — 2y — 2z = O that lies above the triangle boundecTh the es x = 2, y = 0, and y = 3x In the xy-plane. The part of the plane z=6+5x+2y that lies above the rectangle [0,9] x [1,4]. In the zr-plane we have. A)625π B)1250π C)2500π D)1875π 26) 27)Find the volume of the region bounded below by the xy-plane, laterally by the cylinder r = 7 sin θ, and above by the plane z = 10 - x. 4) Setup a double integral that represents the surface area of the part of the surface z = 3xy that lies within the cylinder 5) Setup a double integral that represents the surface area of the part of the surface that lies inside the paraboloid 6) Setup a double integral that represents the surface area of the surface:. This is the equation of the sphere. A(S) = ZZ D s 1 + @z @x 2 + @z @y 2 dA 1. (c) Find an equation for the plane tangent to the surface at (x. For more videos like this one, please. Z 2 0 Z 4 4 (xy +3)dxdyFQ. Find the area of the portion of the unit sphere that is cut out. 1 x 2y 3 0 2 24. Find teh flux of the vector field F =1i + 4j+ 2k across the surface S. Problem 3 Find the surface area of the part of z = 1−x2 −y2 that lies above the xy-plane. Find the area underneath the curve `y = x^2+ 2` from `x = 1` to `x = 2`. What do you mean by "depth of penetration"? 13. Formula for surface area. Example The equation z = 3 describes a plane that is parallel to the xy-plane, and is 3 units \above" it; that is, it lies 3 units along the positive z-axis from the xy-plane. Find the surface area of the part of the plane 5x + 5y + z = 25 that lies above the triangle formed by the three points (4,2,0)(8,2,0)(8,6,0) - 7169673. You may use that the surface area of a cone of height hand base radius ris given by ˇr(r+ p h2 + r2). 21 is revolved about the x-axis, it generates a surface of revolution with the property that all light rays L parallel to the x-axis striking the surface are reﬂected to a single point O (the origin). (Orient C to be counterclockwise when viewed from above. Find the circulation of F~(x,y,z) = hy +x2,x,3xz +yi around C, the curve of intersection of the cylinder x2 +y2 = 1 and the plane z = x+2y with counterclockwise orientation when viewed from above. (f) Find the volume of the region inside the cylinder r = asinθ which is bounded above by the sphere x 2 +y +z 2 = a and below by the upper half of the ellipsoid x 2 a 2 +. Your Test will be a mix of multiple choice questions and open answer questions. The part of the plane $ x + 2y + 3z = 1 $ that lies inside the cylinder $ x^2 + y^2 = 3 $ Answer Area of part of the plane that lies inside the given cylinder is $\pi \sqrt{14}$. In the rst octant it lies over a rectanglular region R = f(x;y)j0 x 4; 0 y 5g. Math 114 Practice Problems for Test 2 Comments: 1. Show Solution Okay we've got a couple of things to do here. Suppose that the temperature on this sphere is given. Since we’re given the center of the sphere in the question, we can plug it into the equation of the sphere immediately. The stay-at-home order is reminiscent of Anne Frank stuck inside the Secret Annex. Here is a review exercise before the ﬁnal quiz. point P given by spherical coordinates (4, 317/4, 277/3). The top of the cylinder lies. The part of the plane 3x + 2y + z = 6 that lies in the first octant. Midterm Exam I, Calculus III, Sample B 1. Here we want to find the surface area of the surface given by z = f (x,y) is a point from the region D. We can also write it as. Plane: 10x+3y+z=10 Cylinder: (x^2)/81+(y^2)/100 = 1. x2 + y2 = z2 is the equation of a circular cone, hence the curve lies on a circular cone. Find the volume V of the solid bounded by the right circular cylinder x 2 + y = 1, the ry-plane, and the plane x + z = 1. After computing, we re-derive the area formula. Math 232 Practice Exam #3 Solutions 1. Find the area of the part of the surface z = 1 + 3x + 2y^2 that lies above the triangle? in the x-y plane with vertices (0; 0), (0; 1) and (2; 1). Show Solution Okay we've got a couple of things to do here. if you give your honest and detailed thoughts then people will find new books that are. Answer: We either must have x = 0 which is the yz-plane, or y = 0 which is the xz-plane. (a) Find the volume of the region inside the cylinder x 2+ y = 9, lying above the xy-plane, and below the plane z = y +3. 2 Evaluation of double integrals. Example The equation z = 3 describes a plane that is parallel to the xy-plane, and is 3 units \above" it; that is, it lies 3 units along the positive z-axis from the xy-plane. Find the area of the surface. MA261-A Calculus III 2006 Fall Homework 11 Solutions Due 11/20/2006 8:00AM 12. The part of the plane 3x + 2y + z = 6 that lies in the first octant. Now the plane intersects this cylinder at an angle. The intersection between the sphere and the cylinder in the upper half-space can be. Now, if we look at the picture, the radius is given by 1−x2, so V = Z 1 −1 πr2dx = Z 1 −1 π(1−x2)2dx = π Z 1 −1 1−2x2 +x4 dx = π x− 2 3 x3 + x5 5 1 −1 = π 1− 2 3 + 1 5 −. The cross sections of the solid perpendicular to the y-axis are squares. Find the equations of the tangents to the circle x2 + y2 = 58 that pass through the point (10, 4). Find the area of the region within both circles r = cosθ and r = sinθ. Connect with experts in more than 300 skills and subjects. Find the surface area of the part of the paraboloid z=16-x^2-y^2 that lies above the xy plane (see the figure below). Find the volume of the solid enclosed by the paraboloids z= x2+y2 and z. 625 square feet. Use graphing software to graph the functions specified in Exercises 31–36. In cylindrical coordinates, the solid is de ned by 0 z 1 + r2 (since z= 0 de nes the x-y-plane), and r= p 5 (since ris always positive). Computethedoubleintegral Z 1 0 Z ex x xy2 dydx Problem 14. Find the area of the triangle cut from the first quadrant by the tangent at (1, 1) to 32 + 3y2 + 8x + 16y = 30. Find the training resources you need for all your activities. sheet A in the x = –2. The part of the plane $ x + 2y + 3z = 1 $ that lies inside the cylinder $ x^2 + y^2 = 3 $ Answer Area of part of the plane that lies inside the given cylinder is $\pi \sqrt{14}$. Find the total power passing through a square area of side 25 mm, in the z=0 plane. Thus we only need to compute ¯z The top surface is x2 + y2 + z2 = 2z ⇒ ρ2 = 2ρcos(φ) or ρ = 2cos(φ). Surface Area: In cases where we can write {eq}z = f(x,y) {/eq}, we can write the. y x z Figure1. Simplifyyouranswer. z)k where C is the boundary of the part of the plane 3x + 2y+z = 1 in the ﬁrst octant 10 F(x,y,z) = xyi+2zj+3yk where C is the curve of intersection of the plane x +z = 5 and the cylinder x2 +y2 = 9 12(a)Use Stokes’ Theorem to evaluate R C Fdrwhere F(x,y,z) = x2yi+ 1 3 x 3j+xykand C is the. Answered by Penny Nom. Solution: Let S1 be the part of the paraboloid z = x2 + y2 that lies below the plane z = 4, and let S2 be the disk x2 +y2 ≤ 4, z = 4. We have x2 + y2 + z2 2ax =)z2 = u2 ax. Thus F~N~ = 0 on this part of the surface. Live Music Archive. Your Test will be a mix of multiple choice questions and open answer questions. 0 ≤ y ≤ 2 − 2 x. We can also write it as. Set up a triple integral in spherical coordinates that gives the volume of the solid that lies outside the cone z = 3x2 +3y2 and inside the hemisphere z =. Find the area of the surface. (a) Find the surface area of S. Find the area of the part of the surface z= x2 + 2ythat lies above the triangle with vertices (0;0), (1;0), and (1;2). Find the surface area of the paraboloid z=x^2+y^2 below the plane z=10 Find the surface area of the part of the paraboloid. these regions: 1. Find the surface area of the part of the plane 5x + 5y + z = 25 that lies above the triangle formed by the three points (4,2,0)(8,2,0)(8,6,0) - 7169673. that lies inside the cylinder. Evaluate Z 1 1 Z ˇ 0 x2. ) could be evaluated directly, however, it’s easier to use Stokes’ Theorem. 6 5 Compute the area of the portion of the conical surface x2 + y2 = z2 which lies above the xy-plane and is cut o by the sphere x2 + y2 + z2 = 2ax. Find the ﬂux of → F = x,y2,z upward through the ﬁrst-octant part S of the cylindrical surface x2 +z2 = a2 for 0 6 y 6 b. Find the area enclosed by the given curve, the x-axis, and the given ordinates. (b – 5 pts) Find an iterated integral which computes the volume of the solid which lies below the cone z = p x2 +y2, above the plane z = 0, and inside the sphere x2 +y2 +z2 = 1. The base of a solid is the region in the xy-plane bounded by the lines y = 2x and x = 1. in segment form. Example The equation z = 3 describes a plane that is parallel to the xy-plane, and is 3 units \above" it; that is, it lies 3 units along the positive z-axis from the xy-plane. These problems are taken from old nal examinations. Be sure to specify the domain. Evaluate RRR D x2dxdydz. Since it is. 16 C) 11475 8 D) 1275 8 31) Find the area of the region specified in polar coordinates. Note that there are two. Answer to Find the surface area of the part of the plane 4x+1y+z=1 that lies inside the cylinder x^2+y^2=9. Find the area of the part of the surface z = 1 + 3x + 2y^2 that lies above the triangle? in the x-y plane with vertices (0; 0), (0; 1) and (2; 1). Graph the equation. Joined Jan 28, 2006 Messages 134. Find the area of the surface. OliverKnill Math 21a,Fall 2011 6 Find the center of the sphere x2 +5x+ y2 −2y +z2 = −1. Suppose that the temperature on this sphere is given. enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2+y. is part of the answer. (b) that part of the elliptical paraboloid x + y 2+ 2z = 4 that lies in front of the plane x = 0. 825cm Answer: The area of the equilateral triangle is 10. The bottom surface is x2 +y2 +z2 = 1 ⇒ ρ. Graph the equation. The cylinder x2 +y2 = x 1. 5 Problem 2E. Can someone help me with this/how to do one like it. Short solutions are provided at the end. Your Test will be a mix of multiple choice questions and open answer questions. Sketch the tangent line at x. Now, if we look at the picture, the radius is given by 1−x2, so V = Z 1 −1 πr2dx = Z 1 −1 π(1−x2)2dx = π Z 1 −1 1−2x2 +x4 dx = π x− 2 3 x3 + x5 5 1 −1 = π 1− 2 3 + 1 5 −. (b)Let Sbe the paraboloid z= x2 +y2, for 0 z 9, oriented downward. Answer: Complete the square 7 Find the set of points P = (x,y,z) in space which satisfy x2 + y2 = 9. $\begingroup$ Find the area of the part of the surface z = 1 + 3x + 2y^2 that lies above the triangle? $\endgroup$ - Steven John Apr 18 '13 at 3:39 1 $\begingroup$ You should use $\LaTeX$ to make your answers more readable $\endgroup$ - Stahl Apr 18 '13 at 4:00. V = ∭ U ρ d ρ d φ d z. Solution: Step 1: Given that the radius is 2cm. 26)Find the volume of the region enclosed by the paraboloids z = x2 + y2 - 4 and z = 46 - x2 - y2. Two planes meet over 3y = 2+y ,y = 1. F(x, y, z) = -y2 i + x j + z2 k C is the curve of intersection of the plane y + z = 2 and the cylinder x2 2+ y = 1. Evaluate Z 1 1 Z ˇ 0 x2. 8 a)Find the distances of P = (12,5,0) to each of the 3. Find the minimum distance from the point (2, —1, l) to the plane Find three numbers whose sum is 9 and whose sum of squares is a minimum. x 2y 8 0 23. Find the area of the surface. The surface can be parametrized as r(x;y) = hx;y;x2 + y2iwhere 1 x 2+ y 2. Find the area of the part of the surface z= x2 + 2ythat lies above the triangle with vertices (0;0), (1;0), and (1;2). For this problem, f_x=-2x and f_y=-2y. Example 2 (a) Find parametric equations for the line through (5,1,0) that is perpendicular to the plane 2x − y + z = 1 A normal vector to the plane is:. Review: Double integral of a scalar function. So, all we have to do is: Find the intersections Determine the length of each diagonal distance Find the volume of. These axes divide the plane into four quadrants, as shown in Figure 7. 1- Find the surface area of the part of the plane 5 x + 2 y + z = 2 that lies inside the cylinder x^{2} + y^{2} = 25. Solution: Step 1: Given that the radius is 2cm. The ﬁrst octant is bounded by three coordinate planes x = 0, y = 0, and z = 0. Unformatted text preview: that lies above the xy-plane. The part of the sphere x² + y² + z² = a² that lies within the cylinder x² + y² = ax and above the xy-plane a²(π - 2). The base of a solid is the region in the xy-plane bounded by the lines y = 2x and x = 1. variables, we get: z=6-2y-3x. My Multiple Integrals course: https://www. Find the volume of the region that lies above the paraboloid z= 2x2 +2y2 and lies below the cone z= 2 p x2 + y2. Tell your tutor when you’d like to meet, and only pay for the time you need. Midterm Exam I, Calculus III, Sample B 1. 13 Use Green's theorem in the plane to show that the circulation of the vector eld F = xy2i + (x2y+ x)j about any smooth curve in the plane is equal to the area. Oct 23, 2006 #1 Find the area of the surface. A shared whiteboard lets you draw, graph. Find the radius the common circle of intersection and show that it is equal to \frac{r_1 r_2}{\sqrt{r_1^{2} + r_2^{2}}}. (8) (C) (D) {0. audio All audio latest This Just In Grateful Dead Netlabels Old Time Radio 78 RPMs and Cylinder Recordings. The bottom surface is x2 +y2 +z2 = 1 ⇒ ρ. See attached file for full problem description. Find the ﬂux of across the part of the cylinder that lies above the -plane and between the planes and with upward orientation. Find the volume of the wedge. The surface you are integrating is the. It is generally believed by mathematicians that π is normal. Find the surface area of the portion of sphere, center (0,0), of radius 4 that lies inside the cylinder x2 +y2 = 12 and above the xy-plane. Evaluate R2 x2 p 4R x2 2 p 4 x R4 2+y xdzdydx. 0 50 The boundary of the surface is the edge which consists of four connected parabolas. Z 2 0 Z 4 4 (xy +3)dxdyFQ. Then dS= p 1 + (f x) 2+ (f y)2dA= p 1 + y2 + xdAso the integral is RR D (x2 + y2) p 1 + y2 + x2 dA. 194 Chapter 9 Applications of Integration 11. Surface: 8x 2+ y + z2 = 9; Plane: z = 1 The trace in the z = 1 plane is the ellipse x2 + y2 8 = 1, shown below. Z ˇ 2 0 e25 e16 2 d = (e25 e16)ˇ 4 10. (f) Find the volume of the region inside the cylinder r = asinθ which is bounded above by the sphere x 2 +y +z 2 = a and below by the upper half of the ellipsoid x 2 a 2 +. Answer: The x-, y-, and z-intercepts of the given plane are 2, 2, and 4. x2 + y2 = z2 is the equation of a circular cone, hence the curve lies on a circular cone. Find the surface area of the part of the plane 3x+ 2y+ z= 6 that lies in the rst octant. Answer: Complete the square 7 Find the set of points P = (x,y,z) in space which satisfy x2 + y2 = 9. Spherical Cap. For more videos like this one, please. ∫ C F ⋅ d s. Question: Find the surface area of that part of the plane 5x + 2y + z = 10 that lies inside the elliptic cylinder (x 2)/25 + (y 2)/64 = 1. The surface area will be the integral of jr x r yjover the possible x. Find the intercepts on the axis of x made by the tangent at (- 5, 12) to x2 + y2 = 169. (c) Find an equation for the plane tangent to the surface at (x. The other four surfaces. As seen in Fig. Describe the following regions in terms of the spherical. xy = (1 − y2 )1/2 x2 dy = 8. variables, we get: z=6-2y-3x. Find the equations of the tangents to the circle 2 2 + 2 y2 - 3 x + 5y - 7 = 0 that are perpendicular to the line x + 2 y + 3 = 0. 10 Describe the set xy = 0. Studyres contains millions of educational documents, questions and answers, notes about the course, tutoring questions, cards and course recommendations that will help you learn and learn. Answer and Explanation:. We first compute: 2 2 curl 1 2y x y z y x z i j k F k 6. Find the area of th&surface x2 — 2y — 2z = O that lies above the triangle boundecTh the es x = 2, y = 0, and y = 3x In the xy-plane. Instead of projecting down to the region in the xy-plane,. These axes divide the plane into four quadrants, as shown in Figure 7. Find the surface area of the paraboloid z=x^2+y^2 below the plane z=10 Find the surface area of the part of the paraboloid. Find the surface area of the surface z = 1+3x+2y2 that lies above the triangle with vertices (0;0), (0;1) and (2;1). Since the plane ABC. My Multiple Integrals course: https://www. Find the cross-sections of the surface 2x 2+ 2y + z2 = 1 in the planes x= k, y= kand z= k. audio All audio latest This Just In Grateful Dead Netlabels Old Time Radio 78 RPMs and Cylinder Recordings. Connect with experts in more than 300 skills and subjects. Find the area of the portion S of the plane x+3y+2z = 6 that lies in the ﬁrst octant. Sketch the tangent line at x. dx 1 + y2 In each of Problems 9 through 20: (a) Find the solution of the given initial value problem in explicit. x2 + y2 + z 2 + 2x + 3 y + 6 + (−2)( x − 2 y + 4 z − 9) = 0 x2 + y2 + z 2 + 7 y − 8z + 24 = 0. 5#3) Find the area of the surface. g -22 is the same as -1 × 22. Find the center and radius of the circle with equation x. (20 pts) Find the surface area of that part of the sphere x2 + y2 + z2-4 that lies inside the paraboloid z x2 + y2. is defined everywhere inside. the right circular cylinder whose base is the disk r = 2cosθ (in the xy-plane) and whose top lies in the plane z = 9 −2x. Sketch the region inside both the cardioid r= 1 cos and the circle r= 1, and nd its area. Restatement of the problem: Find the point A (x,y) on the graph of the parabola, y = x 2 + 1, that minimizes the distance d between the curve and the point B (4,1). 6: Oriented Surfaces and Flux Integrals 903 in + 2yj - + k. The part of the plane. (f) Find the volume of the region inside the cylinder r = asinθ which is bounded above by the sphere x 2 +y +z 2 = a and below by the upper half of the ellipsoid x 2 a 2 +. Surface: 8x 2+ y + z2 = 9; Plane: z = 1 The trace in the z = 1 plane is the ellipse x2 + y2 8 = 1, shown below. The part of the surface z = x2 + y2 that is above the region in the xy-plane given by 0 ≤ x ≤ 1, 0 ≤ y ≤ x2. Z xy x2 + y2 dA Problem 10. Find the minimum distance from the point (2, —1, l) to the plane Find three numbers whose sum is 9 and whose sum of squares is a minimum. Example 5 asks for the maximum value of f when (x, y, z) is restricted to lie on the ellipse. --Lambiam Talk 22:20, 4 May 2006 (UTC). Consider the sphere x 2+ y + z2 = 9 which models an imaginary planet. Find the surface area of the part of the paraboloid z = 9 - x2 - y2 that lies above the plane z = 3. sheet A in the x = –2. Since it passes through the origin, the equation is z= 4x 3y: (iv) We compute the angle using the dot product cos = vw jjvjjjjwjj = 2 p 6 p 10 = 1 p 15: (v) The plane has the equation 5x 2y z= 7 =) 5 2 x+y+ 1 2 z= 7 2 hence a normal vector is (5 2;1; 1 2. Implicit Di erantion. F(x, y, z) = –y2 i + x j + z2 k C is the curve of intersection of the plane y + z = 2 and the cylinder x2 2+ y = 1. Note that the surface will be bounded by an ellipse. 4 Find the volume of the solid in the first octant (x≥0, y≥0, z≥0) bounded by the circular paraboloid z=x2+y2, the cylinder x2+y2=4, and the coordinate planes. The volume of the small boxes illustrates a Riemann sum approximating the volume under the graph of z=f(x,y), shown as a transparent surface. ρ(x,y) = k(x2 +y2) = kr2, m = R π/2 0 R 1 0. Thus F~N~ = 0 on this part of the surface. Thus we're left with something that looks like half of a cylindrical log. and 0 < f (x, y) everywhere else in the domain. HW11: I graded 16. Example: Find a parametric representation of the cylinder x2 + y2 = 9, 0 z 5. about the z-axis. The set is a union of two planes. 0 ≤ y ≤ 2 − 2 x. Recall that if a line has symmetric equations x x 0 a = y y 0 b = z z 0 c; then the line passes. Show Solution Okay we've got a couple of things to do here. Find The Maximum Area Of A Rectangle Inscribed In A Circle Of Radius R. Stewart 16. Problem 3 (16. Find the surface area of the part of the paraboloid z = 9 - x2 - y2 that lies above the plane z = 3. (1) Find the total mass of the solid that lies in the rst octant and is bounded by the sphere x 2 + y 2 + z 2 = 9 with mass density function f(x;y;z) = z 2. audio All audio latest This Just In Grateful Dead Netlabels Old Time Radio 78 RPMs and Cylinder Recordings. May 16, 2011 254 CHAPTER 13 CALCULUS OF VECTOR-VALUED FUNCTIONS (LT CHAPTER 14) Use a computer algebra system to plot the projections onto the xy- and xz-planes of the curve r(t) = t cost,tsin t,t in Exercise 17. ∫ C F ⋅ d s. I think I’ll write in a diary. Learn more about what 1–to–1 lessons can do for you. 6, 37 Find the area of the surface for the part of the plane 3x + 2y + z = 6 that lies in the ﬁrst octant. Sketch the tangent line at x. Calculus Homework Assignment 9 1. For more videos like this one, please. Example 1 Find the surface area of the part of the plane 3x+2y+z = 6. 1)The solid cut from the first octant by the surface z = 9 - x2 - y 1) Find the volume by using polar coordinates. (Orient C to be counterclockwise when viewed from above. Find the area of the surface. MATH 294 SUMMER 1990 PRELIM 1 # 4 294SU90P1Q4. If Dis the unit disk x 2+ y 1 on the xy-plane then this integral is RR D (x2 + y2) dS. vertex 133. To show that the flux across \(S\) is the charge inside the surface divided by constant \(\epsilon_0\), we need two intermediate steps. Find the surface area of the part of the paraboloid z=16-x^2-y^2 that lies above the xy plane (see the figure below). Questions: Find a parameterization for each surface: 1. Find the center and radius of the circle with equation x. ) could be evaluated directly, however, it's easier to use Stokes' Theorem. quiz problem bank quiz problems find all solutions to the following: xy 2y 40 mph wind is blowing due south (270 you are on plane that has bearing of 60 (aka. Question: Find the area of the surface. surface area. The surface can be parametrized as r(x;y) = hx;y;x2 + y2iwhere 1 x 2+ y 2. Find the area of the region within both circles r = cosθ and r = sinθ. F dS where (x, y, '2z) and S is the upper hemisphere of radius 5 centered at the origin with upward orientation. The imaginary part of is (D) y2-x2 the A particle is moving under the action ot a generalised potential 14. Use a double integral to ﬁnd the volume of the solid under the plane z =2x +2y and over the rectangle R = {(x,y):2 x 3, 1 y 3}. So, our surface is a graph of the function f. (b)Let Sbe the paraboloid z= x2 +y2, for 0 z 9, oriented downward. Find the surface area of the hyperbolic paraboloid z= x2 y2 that lies between the cylinders x2 + y 2= 2 and x + y2 = 6: The partial derivatives of zare z x= 2xand z y= 2y, so the surface area is ZZ R p4x2 + 4y2 + 1dA; where Ris the region between the circles x 2+y2 = 2 and x +y2 = 6. and 0 < f (x, y) everywhere else in the domain. The partial derivatives are r x= h1;0;2xiand r y= h0;1;2yi and their cross product is r x r y= h 2x; 2y;1i. 44 square feet. I think I’ll write in a diary. (Orient C to be counterclockwise when viewed from above. The part of the sphere x y z z2 2 2 4 that lies inside the paraboloid z x y22. If Dis the unit disk x 2+ y 1 on the xy-plane then this integral is RR D (x2 + y2) dS. Geometrically, the line integral is the area of a cylinder of. Solution To simplify the calculation, consider the order of integration. The triangular region Tcan be described by T= f(x;y)j0 y 1;0 x yg: Using the previous theorem with f(x;y) = x+ y2 gives A = ZZ T p (1) + (2y)2 + 1dA = Z 1 0 Z y 0 p 4y2 + 2dxdy = Z 1 0 y p 4y2 + 2dy = 1 8 " 2 3 (4y2 + 2. The part of the plane z=6+5x+2y that lies above the rectangle [0,9] x [1,4]. y = (3x2 − 1)/(3 + 2y) 6. Find the area of the triangle cut from the first quadrant by the tangent at (1, 1) to 32 + 3y2 + 8x + 16y = 30. Question: Find the surface area of that part of the plane 5x + 2y + z = 10 that lies inside the elliptic cylinder (x 2)/25 + (y 2)/64 = 1. Example 5 asks for the maximum value of f when (x, y, z) is restricted to lie on the ellipse. In cylindrical coordinates, the solid is de ned by 0 z 1 + r2 (since z= 0 de nes the x-y-plane), and r= p 5 (since ris always positive). This cylinder can be parameterized by R~( ;z) = h3cos ;3sin ;zi for 0 2ˇand 0 z 5. Plane: 10x+3y+z=10 Cylinder: (x^2)/81+(y^2)/100 = 1. Page 129 Solutions to Practice Final Examination 1. So, our surface is a graph of the function f. Surface Area: In cases where we can write {eq}z = f(x,y) {/eq}, we can write the. Find the area of Φ(D). (1 pt) Use Stoke’s Theorem to evaluate Integral_C (F · dr) where F(x,y, z) = x i+y j+3(x^2 +y^2) k and C is the boundary of the part of the paraboloid where z = 9?x^2 ?y^2 which lies above the xy-plane and C is oriented counterclockwise when viewed from. (a) The part of the cone z = p x2 + y2 below the plane z = 3. (d) Find the equation of the plane through A, B and C. Recall that if a line has symmetric equations x x 0 a = y y 0 b = z z 0 c; then the line passes. 2 Evaluation of double integrals. m Find the ux of F = _. Thus we only need to compute ¯z The top surface is x2 + y2 + z2 = 2z ⇒ ρ2 = 2ρcos(φ) or ρ = 2cos(φ). 2x 3y 6 0 29. enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2+y. Similarly when y, or zis constant we get another ellipse. solving this for z to get it as a function of two. In Exercises 19 and 20, let r(t) = sin t,cost,sin t cos2t as shown in Figure 12. Solution: The plane intersects the rst octant in a triangle with vertices (2;0;0), (0;3;0), and 0;0;6 since these are the intercepts with the positive x, y, and z axes respectively. The unit normal vector on the surface above (x_0,y_0) (pointing in the positive z direction) is. y| over the whole domain D will give the surface area. the paraboloid z = x^2+y^2 that lies inside the cylinder x^2+y^2 =1, oriented upward. Solution: Using the divergence theorem, we can convert the given surface integral to a triple. (a) Find the volume of the region inside the cylinder x 2+ y = 9, lying above the xy-plane, and below the plane z = y +3. A shared whiteboard lets you draw, graph. These problems are taken from old nal examinations. The surface area will be the integral of jr x r yjover the possible x. x y z z = 9− 2x r = 2cosθ Solution: ˚ D dV = ˆπ/2 −π/2 ˆ 2cosθ 0 ˆ 9−2x 0 rdzdrdθ Of course, 9 − 2x = 9− 2rcosθ. (8r − 8r5)dr = 16π 3. The equation of the cylinder is ‰x+. Everywhere on the boundary, 1 ( x2 ( y2 = 1 ( 1 = 0. Find the surface area of the portion of sphere, center (0,0), of radius 4 that lies inside the cylinder x2 +y2 = 12 and above the xy-plane. Stewart 16. The surface is a lled ellipse. Your Test will be a mix of multiple choice questions and open answer questions. Note: The mass of an object is equal to the integral over the object of the density function. HW #1: DUE MONDAY, FEBRUARY 4, 2013 1. The part of the sphere x y z2 2 2 4 that lies above the plane z 1. 2017-10-16: From Paulina: Find the rate of change of the distance between the origin and a moving point on the graph of y=x^2 +1 if dx/dt=2 centimeters per second Answered by Penny Nom. Winter 2012 Math 255 Problem Set 11 Solutions 1) Di erentiate the two quantities with respect to time, use the chain rule and then the rigid body equations. Solution Notice that this is the equation of a circle, even though the coeﬃcients of x2 and of y2 are not equal to 1. We have step-by-step solutions for your textbooks written by Bartleby experts!. ) is written as y = 2 – 2x. Use symmetry to evaluate ZZ R sinxdAfor R= [0;2ˇ] [0;2ˇ]. The volume is computed over the region D defined by 0≤x≤2 and 0≤y≤1. For problems 12-13, nd an equation of the trace of the surface in the indicated plane. Choose the direction of the tangent T to ? S and the normal N to the surface so that the vector N ? T points into the surface S.