# Math Help - Polar coordinates problem

1. ## Polar coordinates problem

Using polar coordinates, how to evaluate the integral where R is the region ?
Thanks for helping!

2. Hello,
Originally Posted by iwonder
Using polar coordinates, how to evaluate the integral where R is the region ?
Thanks for helping!
The same way :

$x=r \cos \theta$
$y=r \sin \theta$

Then $9 \leq x^2+y^2 \leq 49$ can be rewritten as : $9 \leq r^2 \leq 49$
Since r is positive, this is like $3 \leq r \leq 7$

And $\theta \in [0,2 \pi]$, in order to cover all the possibilities.

So you have $\int_0^{2 \pi} \int_3^7 \sin(r^2) ~ r ~ dr ~ d \theta$

$=\left(\int_0^{2 \pi} ~ d \theta\right) \cdot \left( \int_3^7 \sin(r^2) ~ r ~ dr \right)$

make the substitution $u=r^2$ in the second.