Suppose that initially I have a right triangle in the $xy$ plane given by the points $a=(0,0)$; $b=(0,B)$ and $c=(B,B)$, such as shown below:
Since from now on, I will work with polar coordinates, I know that the are of this region can be evaluated as: $$A=\int_{0}^{\pi/4}{\int_{0}^{B/cos\theta}{rdrd\theta}}=\frac{B^2}{2}$$ Which is the same as the regular area of the triangle $A=bh/2$ just in polar coordinates. Now suppose that I have a solid of height $H$ where the face $bc$ is given by any generic function $z=f(r,\theta)$ (in cylindrical coordinates), while the others two faces ($ab$ and $ac$) are planes parallel to the $z$ axis. For instance, a solid like this can be visualized as the one below:
My question is on how can I write the equations for volume and surface area of this solid as a function of $f(r,\theta)$? So that I can evaluate these quantities for any $f(r,\theta)$. Thank you in advance!
Edit: Suppose for example that this figure is 1/8 of a pyramid trunk with a square base. How can I prove the volume and surface equations for the pyramid adopting cylindrical coordinates (similar to what I did with the triangle)?