Sketch the region and find the enclosed area.
i) The sketch of the region
ii) Area =83
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=x2 and y=4x-x2
The point of intersection occurs when both the equation are equal to each other, that is,
Thus, the points of intersection is at x=0 and x=2. The region is sketched in the following figure.
Here,4x-x2≥x2 when 0≤x≤2. Therefore, let’s assume that
Therefore, the required area is
After simplifying this, it becomes
Using the sum and difference rule and the constant multiple rule of integral, it becomes
Compute the integral using the standard integration rule.
Evaluate the integral by plugging the upper and the lower limits of integration.