Using Polar Coordinates to Calculate Double Integral

I've encountered a homework problem which has had me a little bit confused, as it doesn't feel as though it marries up very closely with the content in my textbook. I guess that what I'd appreciate most is if someone could verify that I have answered this correctly, and potentially clarify any errors that I've made.

Here is the problem statement:

Let $S$ define the region inside the cardioid $r=1+\sin(\theta)$, above the x-axis. Using polar coordinates, evaluate $\iint_{S}\frac{1}{\sqrt{x^2+y^2}}dA$.

Here's my attempt at the problem:

Defining $S$ as $S=\{(r,\theta)\ |\ 0\le\theta\le\pi,\ 0\le r\le1+\sin(\theta)\}$,

$\iint_{S}\frac{1}{\sqrt{x^2+y^2}}dA = \int_0^\pi \! \int_0^{1+\sin(\theta)} \frac{1}{r}r\ dr\ d\theta = \int_0^\pi \! \int_0^{1+\sin(\theta)} 1\ dr\ d\theta=\pi+2.$

Thank you very much!

• looks alright ! – Mathematician May 16 '13 at 8:10
• When you just write out function names like that, $\TeX$ interprets that as a juxtaposition of variable names and formats it accordingly. To get the appropriate font and spacing, you can use predefined commands like \sin, or, if you need an operator name for which there isn't a predefined command, you can use \operatorname{name}. – joriki May 16 '13 at 8:25
• @joriki - Thanks! I'm a bit new to this. :-) – drokkin May 16 '13 at 8:28
• @Lester: There's an edit link underneath the question. – joriki May 16 '13 at 13:24