Area inside $r = 1 + \cos \alpha $

Question

We are studying polar equations.

Calculate the surface inside $r = 1 + \cos \alpha $ and outside $r = 1$.

I know the area inside $r = 1 + \cos \alpha $ being $\frac{3 \pi}2$ because I calculated $\int_0^{2\pi} 1 + \cos(\alpha) \,\mathrm{d}\alpha$

The given solution is $2+\frac{\pi}4$

How can I visualize this integral? Which formula I should use to obtain the answer and why?

A formula we are given is $$S =\int_{\alpha_0}^{\alpha_1}\frac{r^2(\alpha)}{2}d\alpha~.$$

Answer 1

Hint:

The area of a rectangle of height $y$ and basis $dx$ is $y\,dx$.

The area of a circular sector of radius $r$ and aperture $d\alpha$ is $\dfrac{r^2}2\,d\alpha$.