To sketch and find:
Sketch the region and find the enclosed area
i) The sketch of the region
ii) Area =2-π2
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=cosx and y=1-2xπ
From the graph, the points of intersection are at x=0, x=π2 and x=π. The region is sketched in the following figure.
The shaded region is a symmetric about x=π2
Where A1 is the area under the curve from x=0 and x=π2
Here,cosx≥1-2xπ when 0≤x≤π2. Therefore, let’s assume that
Therefore, the required area is
Compute the integral using the standard integration rules.
Evaluate the integral by plugging the upper and the lower limits of integration.
So, from this, A is given by