The setup for this problem is more difficult than the actual integration.
Since the problem statement calls for integration over y, an expression for x in terms of y must be developed.
At first sight this is easy: solve the first equation (the curve function) for x . This results in
x = 2 sin(y). It is also easy to see that the limits on the integral should be 0 to pi/2.
However, a straightforward integration of 2 sin(y) dy does not give the area of the desired region. Instead, it gives the area of a region bounded by the curve, the y axis and the line y =pi/2.
The area for the region of the question is the difference between [the area of the rectangle bounded by 4 lines:
the y axis, the line y =pi/2 , the x axis and the line x 2] and [the integral of 2 sin(y) dy]. In other words, the desired area is found by a method of subtraction. The area of the rectangle (LxW) is 2 pi/2 = pi , so
Desired area = pi - ∫ 2 sin(y) dy {limits 0 to pi/2}
The anti-derivative of sin is - cos so
Desired area = pi - [ -2 cos(y) ] | 0 pi/2
= pi - 2
