Question

Problem 8

Answer 1

Given information

Semicircle with radius and horizontal diameter , as well as tangent lines at and .

Step-by-step explanation

Step 1.

We use to shift the origin to the semicircle's center as the -axis, and a vertical line through the semicircle's center as the -axis.

The shadow region has now been separated into four pieces , as shown in the picture.

Note that,  and

As a result, the shadow region's area

               

So,  

Differentiate with respect to the shadow area to reduce it.