# Double integral from Cartesian to polar coordinates

I want to write the following double integral

$$\int_0^2 \int_{0}^{\sqrt{1-(x-1)^2}} \frac{x+y}{x^2+y^2} \, \mathrm d x \mathrm d y$$

in polar coordinates. The region is a circle centered at $$(1,0)$$ and with radius $$1$$.

I am having problems with finding the bounds for $$r$$. When we are on the circle,

$$(r \cos\theta -1 )^2 + r^2 \sin^2 \theta = 1$$

which implies that $$r^2 -2 r \cos \theta +1 =0$$. At his point I am stuck. Can you please explain how to find the polar equivalent of this integral? Thank you.

• Your bounds for $r$ are allowed to be a function of $\theta$, so can you just use the quadratic formula to solve for $r$? What does the give you? – Jeremy Dover Apr 7 at 17:25
• What about centering the circle? $x'=x-1$ – Wilem2 Apr 7 at 17:50

Guide:

$$\theta$$ goes from $$-\frac{\pi}2$$ to $$\frac{\pi}2$$.

We have $$r^2-2r\cos \theta = 0$$

(You forgot to cancel out the $$1$$ on both sides).

That is the upper limit of $$r$$ is $$r= 2\cos \theta$$.

That is

$$\int_{0}^\frac{\pi}{2}\int_0^{2\cos \theta} \frac{r\cos \theta + r\sin \theta}{r^2}\cdot r\, \, drd\theta$$

• Thank you. By the way, the lower limit of $\theta$ must be $0$, right? – Ninja Apr 7 at 18:42
• ah, yes, you are right, we are integrating only the upper disc. – Siong Thye Goh Apr 7 at 18:47