find the area enclosed by $ f(x)=x+\sin(x)$ and its inverse from $x=0$ to $x=2$

Question

I don't have a single clue to start, and we cant find the inverse so we must use some properties, but which ones?

thanks

Answer 1

Before we dive into it, I'd just like to point out: a function and its inverse almost always intersect on the line $x=y$.

We'll be finding the area in this region. Note that the two functions intersect at $(\pi,\pi)$ so we want:

$$2\int_0^\pi \int_x^{x+\sin(x)} \mathrm dy\mathrm dx$$

which is half of the area (the top half), doubled. We evaluate:

$$2\int_0^\pi \sin(x)\mathrm dx=-2\cos(x)\biggr\rvert_0^\pi=4$$