Answer:
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)