Exercises. . Evaluate f (, ) xydawhere. 2, 1 x< 3, 0 y< (Answer: 14) 3, 3 x 4, 0 y 2. (Answer: 12) ) f x. ln 2 1 ( ) = (Answer: 1)
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1 Eercises Eercises. Let = {(, : 4, }. Evaluate f (,, <, =, 4, f,, <, < =, <,, 4, a. f (, b. ( dawhere (Answer: 4 (Answer:. Evaluate the following integral a. dd(answer: b. e dd (Answer: c. sin dd (Answer: d. dd (Answer: 4 e. ( g. ( + dd(answer: f. e dd (Answer: ln dd(answer: ln ln h. + e dd (Answer: i. dd (Answer: ln j. + ( ln e dd (Answer: ( ln. Evaluate the double integral over the rectangular region. a. da =, :, (Answer: {( } b. cos( + da {(, : 4 4, 4} c. 4 da {(, :, } = (Answer: = (Answer: 4. Evaluate the following integrals a. dd (Answer: 9 4 b. dd c. (Answer: 9 sin dd (Answer: d. a a cos dd (Answer: a f. dd e. ( + dd(answer: = 5. da, where is the region bounded b =, =, and = (Answer:, where is the region enclosed b,,,. cos da (Answer: = = = and =. mongmara@gmail.com
2 Eercises 7. da where is the region bounded b. ( + =, = and =. (Answer: 57 da, where is the region in the first quadrant, enclosed b = 4 and =. (Answer: 7 9. da + (Answer: ln 4 da, where is a triangular region with vertices ( (. Compute (... sin r asinθ rdrdθ +, is the region enclosed between cosθ drd θ (Answer: (Answer: ( e da, where enclosed b the circle,,, and (,. = and =. (Answer: + = (Answer: ( e 4. da where is the sector in the first quadrant that is bounded b =, = + + and + = 4. (Answer: ln ( dd (Answer: + dd a a dd, ( a > (Answer: 9 ( + + (Answer: + a 4 dd (Answer: ( e dd (Answer: ( e 4 4. sin ( da, is the region bounded b,,. ( + + z dddz.. 4. (Answer: z zddzd (Answer: 7 9 z dddz (Answer: 4 dzdd (Answer: 5 cos = = = (Answer: ( mongmara@gmail.com
3 Eercises 5. Use spherical coordinate to compute (Answer: ( e. Use spherical coordinate to compute (Answer: ( e dzdd + + z z dzdd 7. Use double integral to find the volume of the solid tetrahedron that lies in the first octant that is bounded b the three coordinate planes and the plane z = 5. ( 5. Find the volume of the solid that is bounded above b the plane z = + + below b the -plane and laterall b = and =. (Answer: Use double integral to find the volume of the solid that is bounded above b the paraboloid z = 9 +, below b the plane z = and laterall b the planes z =, =, = and =. (Answer: 7. Use double integral to find the volume of the wedge cut from the clinder 4 + = 9 b the plane z = and z = +. (Answer: 7. Use double integral in the first octant bounded b the three coordinate planes and the plane + = 4 and + 4z = (Answer:. Find the volume of the solid bounded above b the paraboliod z = and below b the -plane. (Answer:. Use double integral to find the volume of the solid common to the clinders + = 5 and + z = 5. (Answer: 4. Find the volume of the solid enclosed b the sphere + + z = 9 and the clinder 4 + =. (Answer: ( 7 5. Volume of the solid that is bounded above b the cone z = +, below b the plane, and laterall b the clinder + =. (Answer: 9. The integral + e Let the value of the integral be I. Thus dwhich arises in probabilit theor, can be evaluated using a trick. + + since the letter used for the variable of integration in a I = e d= e d definite integral does not matter + + ( + a. Show that I e dd = b. Evaluate I b converting to polar coordinate and find I. 7. Find the surface area of the portion of the paraboloid z = + below the plane z =. (Answer: ( 5 5+ mongmara@gmail.com
4 Eercises. Find the surface area of the portion of the cone z = that is above the region in 5 the first quadrant bounded b the line = and parabola =. (Answer: 9. Find the surface area of the portion of the paraboloid z = that is above the plane. (Answer: ( Find the surface area of the portion of the surface z = that is above the sector in the first quadrant bounded b the line =, = and the circle + = 9. (Answer: ( 4. Find the surface area of the portion of the sphere + + z = between the plane z = and z =. (Answer: 4. Find the surface area of the portion of + z = that lies inside the circular clinder + =. (Answer: 4. Compute sin zdv, G is the rectangular bo defined b the G inequalities,, z. (Answer:( 44. Use triple integral to find the volume of the solid in the first octant bounded b the coordinate planes and the plane+ + 4z =. (Answer: Use triple integral to find the volume of the solid bounded b the surface = and planes + z = 4and z =. (Answer: Use triple integral to find the volume of the solid enclosed between the elliptic clinder + 9 = 9and the planes z = and z = +. (Answer:9 47. Use triple integral to find the volume of the solid bounded b the paraboloid z = 4 + and parabolic clinder z = 4. (Answer:. 4. Use triple integral to find the volume of the solid that is enclosed between the sphere + + z = a and the paraboloid az = +. (Answer: ( 7 a. 49. Let G be the tetrahedron in the first octant bounded b the coordinate planes and the z planes + + =, ( a>, b>, c>. a b c a. List si different iterated integrals that represent the volume of G. b. Evaluate an one of the si to show that the volume of G. (Answer: abc. 5. A lamina with densit δ (, = + is bounded b the -ais, the line = and the 9 curve =. Find its mass and center of mass. ( m=,(, =, 7 5. A lamina with densit δ (, = is in the st quadrant and is bounded b the circle + = a and coordinates aes. Find the mass and center of mass. ( a 4,(, a, a m= = A triangular lamina is bounded b = and =, and -ais. Its densit isδ =. Find centroid of the lamina. ( (, =, mongmara@gmail.com 4
5 Eercises 5. A lamina of densit occupies the region above -ais and between the circles 4( b a + = a and + = b ( a< b. Answer: m= ( b a (, =, ( b a 54. A cube is defined b the three inequalities a, a, z a, has densit δ (, z, = a. Find its mass and center of mass. Answer: 4 a a a a m = (,, z =,,. mongmara@gmail.com 5
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