- #1
Telemachus
- 835
- 30
Homework Statement
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:
[tex]\begin{Bmatrix}{ x=r\cos\theta} \\y=r\sin\theta \\z=r\end{matrix}[/tex]
And then I've stated the integral this way:
[tex]\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{r}\displaystyle\int_{r}^{h}rdzdrd\theta=\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{r}r(h-r)drd\theta=\displaystyle\int_{0}^{2\pi}\displaystyle\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)[/tex]
But I should get: [tex]V_{cone}=\displaystyle\frac{\pi r^2 h}{3}[/tex]
I think I'm giving wrong limits for the integration.
Help pls :)
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