Sketch the region and find the enclosed area.
i) The sketch of the region
ii) Area =2215
The area A of the region bounded by the curves x=f(y), x=g(y) and the lines y=c and y=d is
Let’s denote the right boundary by xR and the left boundary by xL, then the area between these curves is given by
x=y4, y=2-x And y=0
Write both the equations as x in the terms of y.
x=2-y2 --- (1)
x=y4 --- (2)
The point of intersection occurs when both the equations are equal to each other, that is,
y=1 and y=-1
Exclude y=-1, because y≥0.
Thus, the points of intersection in the region under consideration is at y=1. The region is sketched in the following figure.
Here, the right curve is x=2-y2 and the left curve is x=y4. Therefore, let’s assume that
Therefore, the required area is
Compute the integral using the standard integration rule.
Evaluate the integral by plugging the upper and the lower limits of integration.