Use polar coordinates to find the volume of the solid inside the cylinder $x ^ { 2 } + y ^ { 2 } = 16$ and the ellipsoid $2 x ^ { 2 } + 2 y ^ { 2 } + z ^ { 2 } = 64$.
A) $943.21$
B) $843.21$
C) $1,043.21$
D) $793.21$
E) $693.21$

Question

Quizplus · Accepted Answer

The volume is calculated by setting up the integral in polar coordinates. The limits for $r$ are from 0 to 4, and for $\theta$ from 0 to $2\pi$. The height $z$ is given by the ellipsoid equation $z = \sqrt{64 - 2r^2}$. The integral evaluates to $\frac{256\pi}{3}$, which is approximately 693.21.

Use Polar Coordinates to Find the Volume of the Solid x2+y2=16x ^ { 2 } + y ^ { 2 } = 16x2+y2=16

Use Polar Coordinates to Find the Volume of the Solid $x ^ { 2 } + y ^ { 2 } = 16$