#### To determine

**To use:**

A computer algebra system to find the exact area enclosed by the two curves.

#### Answer

Area = 126-9

#### Explanation

1) **Concept:**

Graph the two curves and set up the integral for the area using the concept of area between two curves.

Area between two curves:

The area between the curves y=fx and y=gx is ∫ab fx-g(x) dx

**2) Given:**

The equation of the curves y=x5-6x3+4x and y=x

**3) Calculation:**

Graph the functions y1=x5-6x3+4x and y2=x

The graph is shown below.

The curve intersects in four places resulting in four distinct regions.

The graph is symmetric about the origin since f(x) and g(x) are the odd functions.

Find the intersection points by equating two equations.

x5-6x3+4x = x

x5-6x3+3x =0

Factoring x, x(x4-6x2+3) =0

x2=0=>x=0

Solve (x4-6x2+3) =0 by using the quadratic formula.

x2=6±36-122 =3±6

that is, x= -3+6, -3-6, 0 , 3-6, 3+6

The exact area is

2∫23+6fx-g(x) dx

2∫03-6x5-6x3+4x-x dx+ 2∫3-63+6x-x5-6x3+4xdx

By using a computer algebra system

Compute the integral using Mathematica.

Use the command

Integrate2x5-6x3+4x-2x,x,0,3-6

The output is: -9+46

Then use the command

Integrate2x-2x5-6x3+4x,x,3-6,3+6

Output is 86

Add the two outputs.

Area = 126-9

**Conclusion:**

Area = 126-9