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