problem with logic of area of circle proof

Question

I tried proving the formula for area of circle, what I did was talk a circle of r radius and find the area under the curve that is $\int_{0}^{r}\int_{0}^{r} \sqrt{x^2+y^2} dxdy$ and then converting it to polar coordinate and solving it, which will give $\frac{1}{4}^{th}$ of the total area, why is this approach wrong?

Answer 1

As pointed out in the comment, the mistake lies in your thinking that integrating $\sqrt{x^2+y^2}$ over the surface gives you the area, which is wrong. Your integral gives one-fourth of volume of the cone $x^2+y^2=z^2$ in the 3d euclidean octant with height $r\sqrt{2}$. The right integral is $4\int_0^r\int_0^{\sqrt{r^2-y^2}} dx dy$

Answer 2

If you want to use polar coordinate, consider$$\int_{0}^{2\pi}\int_{0}^{r} r drd\theta$$ which is also the $$\int_{0}^{2\pi}\int_{0}^{r} \sqrt{x^2+y^2} drd\theta$$ you wanted to integrate. Just the boundaries wrong.