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
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 $\displaystyle z=0$, we see that both equations take on the form of $\displaystyle x^2+y^2=16$
This implies that our solid is bounded as so:
In Rectangular: $\displaystyle -4\leq x\leq 4, \ \ -\sqrt{16-x^2}\leq y\leq \sqrt{16-x^2}, \ \ -\sqrt{16-x^2-y^2}\leq z\leq 16-x^2-y^2$
In Cylindrical: $\displaystyle 0\leq r\leq 4, \ \ 0\leq \vartheta\leq 2\pi, \ \ -\sqrt{16-r^2}\leq z\leq 16-r^2$
In Spherical: $\displaystyle 0\leq\vartheta\leq 2\pi, \ \ 0\leq\varphi\leq \pi, \ \ 0\leq \varrho\leq 4$
Hopefully, you can take it from here.
--Chris