Use cylindrical coordinates to find the volume of the solid inside the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64$ and above the upper nappe of the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$.
A) $\frac { 512 } { 3 } ( 2 - \sqrt { 2 } )$
B) $\frac { 8 } { 3 } ( 2 - \sqrt { 2 } ) \pi$
C) $\frac { 512 \sqrt { 2 } } { 3 } \pi$
D) $ 512 ( 2 - \sqrt { 2 } ) \pi$
E) $\frac { 512 } { 3 } ( 2 - \sqrt { 2 } ) \pi$

Question

Quizplus · Accepted Answer

To find the volume of the solid inside the sphere $x^2 + y^2 + z^2 = 64$ and above the upper nappe of the cone $z^2 = x^2 + y^2$ using cylindrical coordinates, we first express the equations in cylindrical coordinates. The sphere becomes $r^2 + z^2 = 64$ and the cone becomes $z^2 = r^2$. The volume can be found by integrating in cylindrical coordinates, where the limits for $r$ are from 0 to 8 (the radius of the sphere), the limits for $\theta$ are from 0 to $2\pi$, and the limits for $z$ are from $r$ (the cone) to $\sqrt{64 - r^2}$ (the sphere). The volume integral in cylindrical coordinates is $\int_{0}^{2\pi} \int_{0}^{8} \int_{r}^{\sqrt{64 - r^2}} r dz dr d\theta$. Solving this integral gives the volume of the solid as $\frac { 512 } { 3 } ( 2 - \sqrt { 2 } ) \pi$, which corresponds to choice E.

Use Cylindrical Coordinates to Find the Volume of the Solid x2+y2+z2=64x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64x2+y2+z2=64

Use Cylindrical Coordinates to Find the Volume of the Solid $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64$