Question

Find the area of the region bounded by the curves y equals the inverse sine of x divided by 4, y = 0, and x = 4 obtained by integrating with respect to y. Your work must include the definite integral and the antiderivative.

Answer 1

why all the words?
y = arcsin(x/4)

using thin horizontal strips, the area is

a = ∫[0,π/2] (4-x) dy
But x = 4siny, so
a = ∫[0,π/2] (4-4siny) dy
= 2(π-2)

You can check your work using vertical strips:

a = ∫[0,4] y dx
= ∫[0,4] arcsin(x/4) dx
= 2(π-2)