Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate.

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1 Math 10 Practice Problems Sec Name Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) x dy dx -5 0 A) 5 B) C) 15 D) 5 ) x (8 + ln 9) A) x dy dx + y B) (8 - ln 9) C) (8 - ln 9) D) (8 + ln 9) ) 16 - y (x + y 5/ ) dx dy y A),68 B) 096 C) ,8 D) Find the area of the region specified in polar coordinates. ) the region enclosed by the curve r = 10 cos A) 100 B) 50 C) 100 D) 5 5) the region enclosed by the curve r = 10 + cos A) B) C) 101 D) 6 6) the region inside both r = 8 sin and r = 8 cos A) 8( - 1) B) 16( - 1) C) 8( - ) D) 16( - ) ) the region inside r = 16 sin and outside r = 8 A) 6 + B) 6 + C) 6 + D) 6 + Evaluate the integral. 8) y z yz dx dz dy 0 5 A) 68 B) - 18 C) D) - 1

2 Find the volume of the indicated region. 9) the tetrahedron cut off from the first octant by the plane x 8 + y + z = 1 A) 11 B) C) 11 D) 56 10) the region bounded by the paraboloid z = x - y and the xy-plane A) 500 B) 10,000 C) 1000 D) ) the region common to the interiors of the cylinders x + y = 6 and x + z = 6 A) 86 B) 88 C) 115 D) 1 Evaluate the integral by changing the order of integration in an appropriate way. 1) 1 x sin z z 0 0 y A) 1 - cos B) 1 - sin C) 1 + cos D) 1 + sin 1) e -(x + z) dx dy dz 0 0 y/9 A) e-110 B) e-100 C) e-100 D) e-110 Solve the problem. 1) Set up the triple integral for the volume of the sphere = in spherical coordinates. A) sin d d d B) d d d C) / sin d d d D) / d d d 15) Set up the triple integral for the volume of the sphere = in cylindrical coordinates. A) 16 - r dz dr d B) 16 - r r dz dr d r r C) 16 - r dz dr d D) 16 - r r dz dr d

3 16) Set up the triple integral for the volume of the sphere = 8 in rectangular coordinates. A) x 6 - x - y x B) 6 - x - y x x - y C) x 6 - x - y x D) x x - y x x - y 1) Set up the triple integral for the volume of the sphere = 9 in rectangular coordinates x A) x - y x B) 81 - x 8 - y x C) x 81 - x - y D) x 81 - x - y x x - y x x - y Provide an appropriate response. 18) What form do planes perpendicular to the z-axis have in spherical coordinates? A) = a sin B) = a cos C) = a sec D) = a csc 19) What form do planes perpendicular to the y-axis have in cylindrical coordinates? A) r = a cos B) r = a sec C) r = a sin D) r = a csc 0) What form do planes perpendicular to the y-axis have in spherical coordinates? A) = a csc B) = a csc C) = a sin sin D) = a csc csc 1) What does the graph of the equation = look like? A) A plane B) A cone C) A line D) A cylinder ) What does the graph of the equation = look like? A) The xz-plane B) The z-axis C) The y-axis D) The xy-plane ) What does the graph of the equation = 0 look like? A) The xy-plane B) The y-axis C) The yz-plane D) The z-axis ) What does the graph of the equation = 0 look like? A) The yz-plane B) The xz-plane C) The x-axis D) The y-axis 5) What does the graph of the equation = sec look like? A) A cylinder B) A plane C) A line D) A sphere

4 Evaluate the cylindrical coordinate integral. 6) 8 8r dz r dr d 0 0 r A) 8 B) 8,6 C),68 D) 58 ) / 10 1/r cos dz r dr d 0 5 1/r A) 10 - ln B) 5 + ln C) 5 - ln D) 10 + ln 8) 5 r 10 dz r dr d 0 r A) 0 B) 5 C) 105 D) 105 Solve the problem. 9) Find the mass of the solid in the first octant between the spheres x + y + z = 81 and x + y + z = 100 if the density at any point is inversely proportional to its distance from the origin. A) 8k B) k C) k D) 6 k Use cylindrical coordinates to find the volume of the indicated region. 0) the region enclosed by the paraboloids z = x + y - and z = - x - y A) 185 B) 65 C) 500 D) 150 1) the region enclosed by the paraboloids z = x + y - 6 and z = 1 - x - y A) 1,88 B) 096 C) 819 D) 16,8 Use spherical coordinates to find the volume of the indicated region. ) the region bounded above by the sphere x + y + z = 6 and below by the cone z = x + y A) 18( - ) B) ( - ) C) ( - ) D) 18( - ) ) the region inside the solid sphere that lies between the cones = and = A) B) 9 C) 9 D) Find the center of mass of the thin rod with the given density function. ) Find the center of mass of a thin rod that lies along the first-quadrant portion of the circle x + y = 6 if the density of the rod is = 6sin. A) x = 0, y = B) x = 6, y = 6 C) x =, y = D) x =, y =

5 5) Find the center of mass of a thin rod that lies along the semicircle y = - x if the density of the rod is = cos. A) x = 0, y = B) x = 0, y = 1 C) x = 1, y = 1 D) x =, y = Solve the problem. 6) Find the center of mass of the rectangular solid of density (x, y, z) = xyz defined by 0 x 9, 0 y 10, 0 z. A) x = 9, y = 5, z = B) x = 6, y = 0 10, z = C) x =, y =, z = 1 D) x = 9, y = 5, z = Find the Jacobian for the given transformation. ) x = 5u - v, y = u + 5v A) - B) -1 C) 1 D) Use the given transformation to evaluate the integral. 8) u = x + y, v = -x + y; -x dx dy, R where R is the parallelogram bounded by the lines y = -x + 1, y = -x +, y = x +, y = x + 5 A) - B) - C) D) 9) x = u, y = 9v, z = w; R z dx dy dz, where R is the interior of the ellipsoid x 9 + y 81 + z 16 = 1 A) 6 B) C) D) 8 5 0) u = x + y - z, v = -x + y + z, w = -x + y + z; (x + y - z) dx dy dz, R where R is the parallelepiped bounded by the planes x + y - z =, x + y - z =, -x + y + z = 8, -x + y + z = 10, -x + y + z =, -x + y + z = 10 A) 108 B) 18 C) 8 D) 1 5

6 Answer Key Testname: MATH 10 SEC ) A ) B ) A ) D 5) B 6) C ) D 8) D 9) A 10) D 11) C 1) A 1) B 1) A 15) B 16) C 1) C 18) C 19) D 0) D 1) B ) D ) D ) B 5) B 6) D ) C 8) C 9) C 0) B 1) B ) C ) B ) D 5) B 6) B ) D 8) C 9) C 0) D 6