Compute area between $y=x$, $x=2$ and x axis, but using Polar coordinates.(without double integral).

Question

Problem: compute area between $y=x$, $x=2$ and $x$ axis, but using Polar coordinates.(without double integral). My idea is to first calculate "normal" type of integral. $$\int_0^2x dx=2$$ Secondly I know that boundaries of new integral are $\pi/4$ (because of $y=x$) and 0 (because of $x$ axis). Also I know that if I want to switch to polar coordinates than $x=r\cos(\alpha)$. But I don't know how to end this problem.

Answer 1

Consider the polar equation of the vertical leg $\rho(\theta)=\frac{2}{\cos(\theta)}$ and use the formula: $$\frac{1}{2}\int_{\theta=0}^{\pi/4}\rho^2(\theta) \ d\theta=\frac{1}{2}\int_{\theta=0}^{\pi/4}\frac{4}{\cos^2(\theta)}(\theta) \ d\theta= 2\left[\tan(\theta)\right]_0^{\pi/4}=2.$$ Quite a complicated way to compute the area of a right triangle!

P.S. See this webpage for more details about hoew to calculate areas with polar coordinates.