Hello! I need assistance with math. Thank you a lot

Question

Find the average value of , on its domain.

Options:

1)	2π
2)	4π
3)	2
4)	4

Answer 1

The domain of y = √(64 - x²) is [-8,8].

Average value = [1 / (8-(-8)] ∫_{(-8 to 8)}√(64 - x²) dx

= (1/16) ∫_{(-8 to 8 )} √(64 - x²)dx

The graph of y = √(64 - x²) is the upper half of the semicircle centered at (0,0) with radius 8.

The value of the integral is then the area of this semicircle.

So, average value = (1/16) [(1/2)π(8)²] = 2π