I have a conceptual question.
Suppose I am given a function to integrate over a unit sphere $\mathbb S^2,$ i.e., $$\iiint_{\mathbb S^2} f(x,y,z) \, dV.$$
When I transfer it to spherical coordinates, I know that I can follow some convention like $x = r \cos a$, $y=r\sin a \cos b$ and $z=r\sin a \sin b.$ (I use these variables.)
Now my question is this: do I get the same answer if I use the limits
- $0 \leq r \leq 1,$ $0 \leq a \leq \pi,$ and $0 \leq b \leq 2 \pi$ and
- $0 \leq r \leq 1,$ $0 \leq a \leq 2 \pi,$ and $0 \leq b \leq \pi?$
I think I will get the same answers, but still, the picture is relatively hard to draw here, so I can't include it, but I will be highly obliged if you can confirm or answer this.
No. Observe that the region described by the first equation is the unit sphere $\mathbb S^2;$ however, the region described by the second equation is certainly not the unit sphere. One can check this by computing the volume of the two objects: we have that $\iiint_{\mathbb S^2} 1 \, dV = \frac{4 \pi}{3};$ however, we have that $$\iiint_\mathcal R 1 \, dV = \int_0^{\pi} \int_0^{2 \pi} \int_0^1 r^2 \sin a \, dr \, da \, db = 0.$$