The question is as follows:
Using spherical polar coordinates, find the volume of the solid specified by R $\leq$ 3 and $0 \leq \theta \leq \frac{\pi}{3} $.
I have two big questions about this problem that got me stuck right at the beginning:
1) What about $\phi$? are there any natural assumptions to make, because I cannot imagine the shape of the cone without it.
2) How would I go about solving this problem, I absolutely do not know where to start? how can I "use spherical polars" Do I equate two different integrals related via a Jacobian over a certain region? How would I find this region.. or am I missing something obvious?
- Wesley
Since limits are not given then one can assume the complete domain for $\phi$
The volume function to integrate is just $1$. Then you need only to solve the integral via the Jacobian of the transformation and integrate in the domain of the solid. The region of integration is given for $R$ and $\theta$. And probably we need to integrate also over the whole domain of $\phi$