# how to solve this double integral with polar coordinates? [closed]

Can i use the polar coordinates to solve this integral?

$$\int_{0}^{2}\int_{x}^{\sqrt{8-x^2}} \frac{1}{5+x^2+y^2}dydx$$

## closed as off-topic by Shailesh, John B, C. Falcon, Juniven, JonMark PerryMar 29 '17 at 4:53

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• Sketch the domain of integration. – Jacky Chong Mar 28 '17 at 23:42
• It seems homework. What course is it for? – Rafa Budría Mar 29 '17 at 19:08

## 1 Answer

Polar coordinates would be a good choice. Here's the drawing of region we are integrating over, It is circular-ish. Note that $\sqrt{8-x^2}=y$ means that $8-x^2=y^2$. Or $x^2+y^2=(\sqrt{8})^2$. So $r=\sqrt{8}$ and $y \geq 0$.

As for $y=x$, in polar coordinates that translates to $\arctan(\frac{y}{x})=\arctan (1)=\frac{\pi}{4}=\theta$. So in polar coordinates we have,

$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \int_{0}^{\sqrt{8}} \frac{r}{5+r^2} dr d\theta$$

• Semi-off topic: which app are you using?? – Stefano Mar 28 '17 at 23:53
• I'm using Desmos @Stefano – Ahmed S. Attaalla Mar 28 '17 at 23:54
• thank you, i was doing that but the limits i used were wrong. And the app he used was desmos. – Emma Wool Mar 28 '17 at 23:55