Evaluate $\iint_{R}(x^2+y^2)dxdy$ $$\iint_{R}(x^2+y^2)dxdy$$ 


  $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$


My attempt :

Jacobian=r

$$=\iint_{R}(x^2+y^2)dxdy$$

$$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$

$$\sqrt{x^2+y^2}=r$$

$$\int_{\theta=\pi/4}^{\theta=3\pi/4}\bigg[\int_{r=0}^{r=2}\bigg(r^2\bigg)dr\bigg]d\theta$$

$$.....=\boxed{\frac{4\pi}{3}}$$


  Is it correct?


>
Multiply Jacobian in integrand

$$\int_{\theta=\pi/4}^{\theta=3\pi/4}\bigg[\int_{r=0}^{r=2}\bigg(r^3\bigg)dr\bigg]d\theta$$
while converting in polar co-ordinate you have directly taken dxdx=drd(theta),
you have to use polar transformation Jacobin to do so, that is dxdx=rdr*d(theta), then integrate normally 

answer will be (ip/2).