How do I change the order of integration?

Question

∫₀¹∫_arcsin(y)^pi/2cos(x)√(cos²(x) +1) dxdy

Answer 1

The limits of integration make a region bounded by x = π/2 and x = arcsin(y) between y = 0 and y = 1.

Now, we note that the intersection between x = arcsin(y) and y = 0 is (0, 0) and between x = arcsin(y) and y = 1 is (π/2, 1).

The region can therefore also be expressed in terms of x first as going from y = 0 to y = sin(x) between x = 0 and x = π/2.

So, our integral becomes

∫₀^pi/2∫₀^sin(x)cos(x)√(cos²(x) +1) dydx