Sketch the region and find the enclosed area.
i) The sketch of the region:
ii) Area =12
The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b is
fx-gx=fx-gx when fx≥g(x)gx-fx when gx≥f(x)
y=x3 and y=x
The point of intersection occurs when both the equation are equal to each other, that is,
x=0 and x2-1=0
x=0 and x2=1
x=0, x=1 and x=-1
Thus, the points of intersection are at x=0, x=1 and x=-1. The region is sketched in the following figure.
The shaded region is a symmetric about x=0
The total area A=2A1, where A1 is the area under the curve from x=0 and x=1.
Here x≥x3 when 0≤x≤1. 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.
So, from this, A is given by