1. ## integral set up..

Find the volume of the solid bounded above by the paraboloid z = 16 - x^2 - y^2 and bounded below by x^2 + y^2 + z^2 =16.

*note, set up the integral in rectangular, cylindrical, and spherical coordinates only. Do not evaluate the integral

2. Originally Posted by boousaf
Find the volume of the solid bounded above by the paraboloid z = 16 - x^2 - y^2 and bounded below by x^2 + y^2 + z^2 =16.

*note, set up the integral in rectangular, cylindrical, and spherical coordinates only. Do not evaluate the integral
Let's do some examining prior to setting up the integration.

At $z=0$, we see that both equations take on the form of $x^2+y^2=16$

This implies that our solid is bounded as so:

In Rectangular: $-4\leq x\leq 4, \ \ -\sqrt{16-x^2}\leq y\leq \sqrt{16-x^2}, \ \ -\sqrt{16-x^2-y^2}\leq z\leq 16-x^2-y^2$

In Cylindrical: $0\leq r\leq 4, \ \ 0\leq \vartheta\leq 2\pi, \ \ -\sqrt{16-r^2}\leq z\leq 16-r^2$

In Spherical: $0\leq\vartheta\leq 2\pi, \ \ 0\leq\varphi\leq \pi, \ \ 0\leq \varrho\leq 4$

Hopefully, you can take it from here.

--Chris