Volume of solid in polar coordinates

Question

I find this kind of questions in a calculus course and I've been trying to figure out what is the sequence of steps to follow in order to solve these.

Consider the following solid, defined in terms of polar coordinates: $0≤r≤R$; $0≤θ≤2π$; $0≤z≤r$. Can you describe this shape? Compute its volume. (There are no equations given, only this constraints.)

I know it is related to the definite integral, but I don't know how to use it without an explicit function for the radius r. I really appreciate it if someone could help me into dealing with this type volume calculations problems.

Answer 1

Hint:

1. The volume of a cylinder of height $R$ and base of radius $R$ is $\pi R^3$.

2. The volume of a cone of height $R$ and base of radius $R$ is $\frac{1}{3}\pi R^3$.

Draw a cylinder of such dimensions and subtract the ice-cream.

Answer 2

The volume of a body $B$ in $n$-dimensional Cartesian coordinate system is $$\iiint\limits_B\mathrm dx_1\,\mathrm dx_2\ldots\mathrm dx_n.$$

Thus, by the change-of-variables theorem, the volume of $B$ in cylindrical coordinate system is $$\iiint\limits_B r\;\mathrm dz\,\mathrm dr\,\mathrm d\theta.$$

So, the given solid has volume $$\int_0^{2\pi}\int_0^R\int_0^rr\;\mathrm dz\,\mathrm dr\,\mathrm d\theta=\frac{2πR^3}3.$$