Write an integral expression for the area of the half-circle, slicing as shown and using *h* as the variable of integration.

Answer

If the slice is *h* below the top of the circle, it's (6 – *h*) above the bottom of the circle:

Let *x* be one side of the useful triangle, so the length of the slice is 2*x*:

The Pythagorean Theorem tells us that

(6 – *h*)^{2} + *x*^{2} = 6^{2},

so

(don't bother simplifying this expression). This means the length of the slice at depth *h* below the top of the circle is

and the area of that slice is

The variable *h* goes from 0 at the top of the half-circle to 6 at the bottom of the half-circle, so the area of the half-circle is