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 2x:

The Pythagorean Theorem tells us that

(6 – h)² + x² = 6²,

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