Compute the following multiple integral:
$$ \int\int_D |2x - y| dx dy $$
where D is the subset of ${\rm I\!R^2}$ consisting of points (x,y) for which:
$$ x^2 + y^2 \leq 1 $$ I assume there could be 2 ways to do this.
1) Integrate over a semicircle and multiple by 2:
$$ 2\int_{-1}^1\int_0^{\sqrt {1-y^2}}|2x-y|dxdy $$
2) Change to polar coordinates:
$$ \int_0^{2\pi}\int_0^1|2rcos(\theta)-rsin(\theta)|rdrd\theta $$
Are either of these right, and how do I actually do them?
hint
the line $y=2x $ intersect the circle at the points
$M_1(x_1=\cos(t),y_1=\sin(t)) $ and
$M_2(x_2=-\cos (t),y_2=-\sin(t) )$
where $t $ is such that
$$\tan (t)=\frac{y}{x}=2$$ and $$\cos(t)=\frac {\sqrt {5}}{5} $$
the integral becomes $$\int_0^1\int_t^{t+\pi}(\sin(u)-2\cos (u))rdrdu+$$ $$\int_0^1\int_{t+\pi}^{t+2\pi}(2\cos (u)-\sin (u))rdrdu$$
You can finish to obtain $$2\sqrt {5} $$