Problems

01

Polar double integrals

Integrate $f$ over $\mathcal{R}$ using polar coordinates:

• (a) $f(x,y)=\sqrt{x^2+y^2}$ where $\mathcal{R}$ is the region with $x^2+y^2\le 2$.
• (b) $f(x,y)=y(x^2+y^2{)}^3$ where $\mathcal{R}$ is the region with $y\ge 0$ and $x^2+y^2\le 1$.
• (c) $f(x,y)=y$ where $\mathcal{R}$ is the region with $x^2+y^2\le 1$ and $(x-1{)}^2+y^2\le 1$.

Sketch the region, change the integral to polar coordinates, and evaluate:

• (d) ${\int }_{-2}^2{\int }_0^{\sqrt{4-x^2}}(x^2+y^2)dydx$.
• (e) ${\int }_0^{1/2}{\int }_{\sqrt{3}x}^{\sqrt{1-x^2}}xdydx$.