To determine

To:

Sketch the region and find the enclosed area.

Answer

i) The sketch of the region:

ii) Area =12

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=x3 and   y=x

3) Calculation:

The point of intersection occurs when both the equation are equal to each other, that is,

x3=x

x3-x=0

x(x2-1)=0

that is,

x=0 and x2-1=0

x=0 and x2=1

x=0, x=1 and   x=-1

Thus, the points of intersection are at x=0,  x=1 and   x=-1. The region is sketched in the following figure.

The shaded region is a symmetric about   x=0

The total area A=2A1, where A1 is the area under the curve from   x=0 and   x=1.

Here x≥x3 when   0≤x≤1. Therefore, let’s assume that

fx=x

gx=x3

Therefore, the required area is

A1=∫01x-(x3)dx

A1=∫01x-x3 dx

Compute the integral using the standard integration rule.

A1=x22-x4401

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

A1=122-144-0-0

A1=12-14

A1=14

So, from this, A is given by

A=2A1

A=214

A=12

Conclusion:

i) The sketch of the region:

ii) Area =12