Sketch the region and find the enclosed area
i) The sketch of the region
ii) Area =34
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=1x2, y=x and y=18x
The point of intersection occurs when both the equations are equal to each other. So intersection of y=1x2 and y=x is given by
Intersection of y=1x2 and y=18x is given by
that is at x=2
Intersection point of y=x and y=18x is given by
that is, x=0
Thus, the points of intersection are at x=2 and x=3 and x=0. The region is sketched in the following figure.
Where A1 is the area between the curves from 0≤x≤1
And A2 is the area between the curves from 1≤x≤2
Between x=0 and x=1 the area is bound by curves y=x and y=18x
Here x≥18x 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.
Now 1x2≥18x when 1≤x≤2. Therefore, let’s assume that
So, from this, A is given by: