Double Integral of function in region bounded by two circles

[songoku]

Homework Statement

Let R be the region which lies inside the circle $x^2+y^2=100$ and outside the circle $x^2-10x+y^2=0$
a) Sketch the two circles and shade the region R in your diagram
b) Evaluate the integral $\iint_R \frac{1}{\sqrt{x^2+y^2}}~dA$

Relevant Equations

$\iint_R f(x,y)~dA=\int_{\alpha}^{\beta} \int^{b}_{a} f(r,\theta) r~dr~d\theta$

$r=2a \cos\theta+2b\sin\theta$

The polar form of $x^2+y^2=100$ is $r=10$ and polar form of $x^2-10x+y^2=0$ is $r=10 \cos\theta$

My idea is to divide the working into two parts:
1) find the integral in 1st quadrant and multiply by 2 to include the region in 4th quadrant
2) find the integral in 2nd quadrant and multiply by 2 to include the region in 3rd quadrant

Integral in 1st quadrant:
$$\int_{0}^{\frac{\pi}{2}} \int^{10}_{10\cos\theta} \frac{1}{r}r~dr~d\theta$$
$$=\int_{0}^{\frac{\pi}{2}} \int^{10}_{10\cos\theta} dr~d\theta$$

Integral in 2nd quadrant:
$$\int_{\frac{\pi}{2}}^{\pi} \int^{10}_{0} dr~d\theta$$

Am I correct? Thanks

 

[anuttarasammyak]

My strategy is integral in red - integral in blue.