Let D be the region bounded below by the xy-plane, above by the sphere x^2 + y^2

Question

Let D be the region bounded below by the xy-plane, above by the sphere x^2 + y^2 + z^2 = 100, and on the sides by the cylinder x^2 + y^2 = 36. Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dr dz dtheta. Integral^2 pi_0 Integral^Squareroot 64_0 Integral^10_0 r dr dz dtheta + Integral^2 pi_0 Integral^6_Squareroot 64 Integral^Squareroot 100-z^2 r dr dz d theta Integral^2 pi_0 Integral^Squareroot 64_0 Integral^6_0 r dr dz dtheta + Integral^2 pi_0 Integral^10_Squareroot 64 Integral^Squareroot 100-z^2 r dr dz d theta Integral^2 pi_0 Integral^Squareroot 64_0 Integral^10_0 r dr dz dtheta + Integral^2 pi_0 Integral^6_Squareroot 64 Integral^Squareroot 36-z^2 r dr dz d theta Integral^2 pi_0 Integral^Squareroot 64_0 Integral^10_0 r dr dz dtheta + Integral^2 pi_0 Integral^6_Squareroot 64 Integral^Squareroot 36-z^2 r dr dz d theta

Explanation / Answer

above xy plane (z=0)

in cylindrical coordinates

x=rcos, y=rsin

x2+y2=r2

x2+y2+z2=100 ,x2+y2=36

=>r2+z2=102 ,r2=62

=>r2+z2=102 ,r=6

=>62+z2=102

=>z=64

0<=<=2,0<=r<=6 ,0<=z<=64

and

r2+z2=102

r=[100-z2]

0<=<=2,0<=r<=[100-z2] ,64<=z<=10

dv =r dr dz d

volume=[0 to 2] [0 to 64] [0 to 6] r dr dz d +[0 to 2] [64to 10] [0 to [100-z2]] r dr dz d

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