#### To determine

**To sketch and find:**

Sketch the region and find the enclosed area

#### Answer

i) The sketch of the region

ii) Area =2-π2

#### Explanation

**1) Concept:**

Formula:

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

A= ∫abfx-gxdx

fx-gx=fx-gx when fx≥g(x)gx-fx when gx≥f(x)

**2) Given:**

y=cosx and y=1-2xπ

**3) Calculation:**

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

A=2A1

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

fx=cosx

gx=1-2xπ

Therefore, the required area is

A1=∫0π2cosx-1-2xπdx

A1=∫0π2cosx-1+2xπ dx

Compute the integral using the standard integration rules.

A1=sinx-x+2π·x220π2

A1=sinx-x+x2π0π2

Evaluate the integral by plugging the upper and the lower limits of integration.

A1=sinπ2-π2+π24π-0-0+0

A1=1-π2+π4

A1=1-π4

So, from this, A is given by

A=2A1

A=21-π4

A=2-π2

**Conclusion:**

i) The sketch of the region

ii) Area =2-π2