Evaluate double integral $\iint_A x^2\,dx\,dy$

Question

Calculate double integral

$\iint_A x^2\,dx\,dy$

Where $$ A = \{(x,y)\in R^2 \| x^2 + y^2 \leq 1\}$$

I try do this with polar coordinates $$0 \leq r \leq 1 $$$$ 0 \leq \theta \leq 2\pi $$ But I failed (I'm getting zero, I'm not sure I've correctly written out the polar coordinates)

My integral $$\int_0^{2\pi}\int_0^1(r\cos\theta)^2r\,dr\,d\theta$$

Answer 1

In polar coordinates we get the double integral

$$\int_0^1\int_0^{2\pi}r^2\cos^2\theta\cdot r\,d\theta\,dr=\int_0^1r^3\,dr\,\left.\frac12(\theta+\cos\theta\sin\theta)\right|_0^{2\pi}=\left.\pi\frac{r^4}4\right|_0^1=\frac\pi4$$

Answer 2

Since $x=r\cos\theta$, $x^2=r^2\cos^2\theta$. So, you have to compute\begin{align}\int_0^{2\pi}\int_0^1r^3\cos^2\theta\,\mathrm dr\,\mathrm d\theta&=\left(\int_0^{2\pi}\cos^2\theta\,\mathrm d\theta\right)\left(\int_0^1r^3\,\mathrm dr\right)\\&=\left(\int_0^{2\pi}\frac{\cos(2\theta)+1}2\theta\,\mathrm d\theta\right)\left(\int_0^1r^3\,\mathrm dr\right).\end{align}Can you take it from here?