Sketch the region and find the enclosed area
i) The sketch of the region
ii) Area =43
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=3x-x2, y=x and x=3
The point of intersection occurs when both the equations are equal to each other, that is when
that is at,
x=0 and x=2
Intersection point of y=x and x=3 is at x=3
Thus, the points of intersection are at x=2 and x=3. The region is sketched in the following figure.
Here, x≥3x-x2 when 2≤x≤3. 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.