#### To determine

**To:**

Sketch the region and find the enclosed area.

#### Answer

i) The sketch of the region.

ii) Area =135

#### 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=x4 and y=2-x

**3) Calculation:**

From the graph, the points of intersection are at x=-1 and x=1. The region is sketched in the following figure.

The shaded region is a symmetric about x=0

A=2A1

where A1 is the area under the curve from x=0 and x=1

When x≥0, 2-|x|=2-x

Here 2-x≥x4 when 0≤x≤1. Therefore let’s assume that

fx=2-x

gx=x4

Therefore, the required area is

A1=∫012-x-x4dx

A1=∫012-x-x4 dx

Compute the integral using the standard integration rule.

A1=2x-x22-x5501

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

A1=(21-122-155-0-0-0

A1=2-12-15

A1=1310

So, from this, A is given by:

A=2A1

A=21310

A=135

**Conclusion:**

i) The sketch of the region

ii) Area =135