Answer:
π − 2
Step-by-step explanation:
Graph of the region:
desmos.com/calculator/pcascl0frf
If we integrate with respect to x:
∫₀² (y − 0) dx
∫₀² sin⁻¹(x/2) dx
But we want to integrate with respect to y. Let's start by finding the new limits of integration.
y = sin⁻¹(x/2), so when x = 0, y = 0. When x = 2, y = π/2.
Next, we need to find x in terms of y.
sin y = x/2
x = 2 sin y
So the integral with respect to y is:
∫₀ᵖⁱ² (2 − x) dy
∫₀ᵖⁱ² (2 − 2 sin y) dy
Integrating:
(2y + 2 cos y + C) |₀ᵖⁱ²
(π + 2 cos (π/2) + C) − (0 + 2 cos 0 + C)
(π + 0 + C) − (0 + 2 + C)
π − 2
