A standard application of integration is to find the area between two curves.
The integration unit is the top function minus the bottom function.
The basic integral is
It should be noted that if top and bottom, or left and right, are reversed, the area is negative.

It is always good to start with a problem where we can find the answer without using integration.

A rectangle, Top(x)=4, Bottom(x)=1, Left = -2, Right =3

A triangle, Top(x)=x, Bottom(x)=-1, Left = -1, Right =3

A circle, Top(x)=sqrt(4-x^2), Bottom(x)=-sqrt(4-x^2), Left=-2, Right=2

One can then look are areas we cannot compute with simple geometry.

Bottom(x)=x^2, Top(x)=4, Left=-2, Right=2

Bottom(x)=x^2, Top(x)=x+2, Left=-1, Right=2

One should also look at cases where the "top" and "bottom" curves cross.