Math Help - triple integral

1. triple integral

Hey.

What's the answer for this:

$\int_{0}^{r}dr \int_{0}^{2\pi}rd\phi \int_{0}^{l}dl \rho (1-r/R)$

2. $\rho l2\pi r^{2}[\frac{1}{2}-\frac{r}{3R}]$

3. Originally Posted by jacek
Hey.

What's the answer for this:

$\int_{0}^{r}dr \int_{0}^{2\pi}rd\phi \int_{0}^{l}dl \rho (1-r/R)$
Actually, that is a very badly posed integral. It is never a good idea to use the same letter to represent the variable of integration and a limit of integration and that is done twice here. Other than that, it's pretty close to trivial. We can "separate" those integrals and do them separately:
$\left(\int_0^r (1- r/R)r dr\right)\left(\int_0^{2\pi} d\phi\right)\left(\int_0^l dl\right)$.

And, of course, $\int_0^{2\pi} d\phi$ and $\int_0^l dl$ are trivial!

$\int_0^r (1- r/R)r dr= \int_0^r rdr- \frac{1}{R}\int_0^r r dr$ should also be easy though not "trivial".