Find the volume bounded by the surface $z = x^2 + y^2$ and $x^2+y^2 = 1$ in the first quadrant. The answer is $\pi/8$ using rectangular and cylindrical coordinates and that is the correct answer, but it is $\pi/24$ when I switched to spherical coordinates, as follows, $$\int_0^{\pi/2} \int_{\pi/4}^{\pi/2} \int_0^{\cot{\phi} \csc{\phi}} \rho^2 \sin{\phi} \, d\rho \, d\phi \, d\theta.$$ In general, how do you figure out the bounds for $\theta, \rho$ and $\phi$? I use the conversion formulas but I'm not sure how to proceed in a general way.
In rectangular and cylindrical coordinates, respectively
$$\int_0^1 \int_0^{1-x^2} x^2 + y^2 \,dy \,dx$$ $$\int_0^{\pi/2} \int_0^1 \int_0^{r^2} r \,dz\,dr\,d\theta$$
$\theta$ is sweeping angle in the $xy$-plane. $\phi$ is the sweeping angle between positive $z$-axis and the $xy$-plane.
Also, how to you finish a paragraph and enter onto the next line when you edit this (I'm new to this site)? Thanks a lot!