Answer to Question #211244 in Calculus for Elorm

Question

Evaluate$\intop\intop$ x²+y² +2

Where Ris bounded by x²+ y² =1

Answer 1

Let us evaluate $\iint\limits_R(x^2+y^2+2)dxdy.$

$\iint\limits_R(x^2+y^2+2)dxdy=$

|$x=r\cos \varphi,\ y=r\sin\varphi,\ dxdy=rdrd\varphi, x^2+y^2=r^2$ |

$=\int\limits_0^{2\pi}d\varphi\int\limits_0^1(r^2+2)rdr=2\pi\frac{1}{2}\int\limits_0^1(r^2+2)d(r^2+2)= \pi\frac{(r^2+2)^2}{2}|_0^1=\pi(\frac{9}{2}-2)=\frac{5\pi}{2}.$