Given $S$ as the solid that is outside the cylinder $x^{2}+y^{2}=1,$ inside the sphere $x^{2}+y^{2}+z^{2}=4,$ and above the $x y$ -plane. Set-up an iterated triple integral in spherical coordinates that is equal to $\iiint_{G} f(x, y, z) d V$ where $f(x, y, z)=x \sqrt{x^{2}+y^{2}+z^{2}}$.
How I approached this problem is that since
$f(x, y, z)=x \sqrt{x^{2}+y^{2}+z^{2}}$ and we know that from rectangular to spherical we have $\rho=\sqrt{x^2+y^2+z^2}$ and $x=\rho sin\phi cos\theta$
I know have $\iiint_{G} f(x, y, z) d V = \iiint_{G} \rho sin\phi cos\theta*\rho^2 sin\phi d\rho d\theta d\phi $
I've also determined the bounds for $\theta = 0$,$ 2\pi$ as well as for $\phi = 0$,$\pi/2$
My problem is determining the bounds for $\rho$ as the inner figure is a cylinder. My guess is that the upper bound of $\rho=2$ but I can't determine the lower bound.
Any help would be appreciated. Thanks.
Best Answer
You are right about the outer bound, the inner bound will have to be the cylinder:
$$x^2+y^2=1 \implies \rho\sin\phi = 1$$
In the $\rho\phi$-plane ($\rho$ being the vertical axis) the region of integration will be shaded area to the left of the green line:
You can see from the photo that $\phi$ doesn't actually make it all the way back to $0$ (the $z$ axis in Cartesian), it stops at
$$2\sin\phi = 1 \implies \phi = \frac{\pi}{6}$$
and $\rho$ stops at
$$\rho\sin \frac{\pi}{2} = 1 \implies \rho = 1$$
From here it's easier to see how to set up the integral in both orders.
$$\int_0^{2\pi}\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\int_{\csc\phi}^{2} f\cdot \rho^2\sin\phi \: d\rho d\phi d\theta$$
$$\int_0^{2\pi}\int_{1}^{2}\int_{\sin^{-1}\left(\frac{1}{\rho}\right)}^{\frac{\pi}{2}} f\cdot \rho^2\sin\phi \: d\phi d\rho d\theta$$