And here's the proof thingy:

Let ff[x]=ƒ(x)

The wanted area won't change if we push it one up. We to this to reduce ff to positive function.

We want S+S1+S3.

(x,y) means point x,y.

But

(rectangular with 45 deg)

so S = INEGRAL - S2.

Now we'll find S1 and S3:

(rectangular with 45 deg)

(rectanguler with 45 deg)

Then