Sketch the region and find the enclosed area.
i) The sketch of the region is
ii) Area =4π
The area A of the region bounded by the curves y=fx, 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)
When cosx=2-cosx, we have that cosx=1.
Thus, the points of intersection are at x=0 and x=2π. The region is sketched in the following figure.
Here 2-cosx≥cosx when 0≤x≤2π. Therefore, let’s set
Therefore, the required area is
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.