Answer
$2\pi$
Work Step by Step
Conversion of polar coordinates and Cartesian coordinates are as follows:
a)$r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$
b) $\tan \theta =\dfrac{y}{x} \implies \theta =\tan^{-1} (\dfrac{y}{x})$
c) $x=r \cos \theta$
d) $y=r \sin \theta$
Here, $(x,y)=(1,1)$
This suggests the integral is as follows: $\int_0^{\frac{\pi}{2}} \int_0^{2} (r^2)r dr d\theta =\int_0^{\frac{\pi}{2}} \int_0^{2} r^3 dr d\theta $
or, $\int_0^{\frac{\pi}{2}} \int_0^{2} \dfrac{r^4}{4} d\theta=[\dfrac{2^4}{4}\theta]_0^{\frac{\pi}{2}}=2\pi$
