Use an appropriate coordinate system to compute the volume of the solid below and above .

Question

Use an appropriate coordinate system to compute the volume of the solid below  and above  .

Quizplus · Accepted Answer

We need to use cylindrical coordinates with the vertical axis passing through the midpoint of the two given surfaces. Then, the equation of the top surface becomes $z=1-(r/2)$ and the equation of the bottom surface becomes $z = -1+(r/2)$. Thus, the limits of $r$ are from $0$ to $2$ and the limits of $	heta$ are from $0$ to $2\pi$, while the limits of $z$ are from $-1+(r/2)$ to $1-(r/2)$. Evaluating the integral using these limits gives us the volume of the solid, which turns out to be $4\pi/3$. The answer choice that matches this result is (C).

Use an Appropriate Coordinate System to Compute the Volume of the Solid