Question

Here is the question

Answer 1

Explanation:

The curve is maybe a bit easier to visualize if you write it as x = 4sin(y). The area is

∫[0,π/2] (4-x) dy

= ∫[0,π/2] (4-4sin(y)) dy = 2π-4

∫4-4sin(y) dy = 4y + 4cos(y)

This is a bit easier than using vertical strips, where the area is

∫[0,4] arcsin(x/4) dx

since integrating arcsin(x/4) involves integration by parts.

u = arcsin(x/4)

du = 1/√(16-x^2) dx

dv = dx

v = x

∫arcsin(x/4) dx

= x arcsin(x/4) - ∫x/√(16-x^2) dx

= x arcsin(x/4) + √(16-x^2)