Question

Evaluate the integral by making a change of coordinates.

Answer 1

\[\int\limits_{-3}^{3}\int\limits_{-\sqrt{9-x ^{2}}}^{\sqrt{9-x ^{2}}}\int\limits_{3-\sqrt{x ^2 + y ^2}}^{\sqrt{9-x ^2 - y ^2}} z dz dy dx\]

Answer 2

Change into cylindrical coordinates using the following:
\[\begin{cases}x^2+y^2=r^2\\x=r\cos\theta\\y=r\sin\theta\\dx~dy~dz=r~dr~d\theta~dz\end{cases}\]
(\(z\) remains \(z\).)

Notice that the limits in the \(x\) and \(y\) directions describe a circle in the \(x-y\) plane with radius 3:
\[x^2+y^2=9~~\iff~~y=\pm\sqrt{9-x^2}\]
The integral in cylindrical coordinates is then
\[\large\int_0^3\int_0^{2\pi}\int_{3-\sqrt{r^2}}^{\sqrt{9-r^2}}rz~dz~d\theta~dr\]