Hi there. I haven't used iterated integrals for a while, and I'm studying some mechanics, the inertia tensor, etc. so I need to use some calculus. And I'm having some trouble with it.

I was trying to find the volume of a cone, and then I've found lots of trouble with such a simple problem.

So I thought of using cylindrical coordinates this way:

$\displaystyle \begin{Bmatrix}{ x=r\cos\theta} \\y=r\sin\theta \\z=r\end{matrix}$

And then I've stated the integral this way:

$\displaystyle \displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^ {r}\displaystyle\int_{r}^{h}rdzdrd\theta=\displays tyle\int_{0}^{2\pi}\displaystyle\int_{0}^{r}r(h-r)drd\theta=\displaystyle\int_{0}^{2\pi}\displayst yle\frac{r^2h}{2}-\displaystyle\frac{r^3}{3}=\pi r^2h-\displaystyle\frac{2\pi\r^3}{3}=\pi r^2(h-\displaystyle\frac{2}{3}r)$

But I should get: $\displaystyle V_{cone}=\displaystyle\frac{\pi r^2 h}{3}$

I think I'm giving wrong limits for the integration.

Help pls