#### To determine

**To find:**

The exact volume of the solid obtained by rotating the region bounded by the given curves about the line y= 2 by using a computer algebra system.

#### Answer

V= 419π2+120π-21015π

#### Explanation

**1) Concept:**

i. If the cross section is a washer with inner radius rin and outer radius rout, then area of washer is obtained by subtracting the area of the inner disk from the area of the outer disk,A=π outer radius2 -π inner radius2

ii. The volume of the solid of revolution about x-axis is

V= ∫abA(x)dx

**2) Given:**

y=x2-2x, y=xcosπx4;about y=2

**3) Calculation:**

Bounded region of the given curves is shown below:

From the above graph

x coordinate of the points of intersection: x=0 and x= 2

The cross section is perpendicular to the x-axis, and it is a washer.

Thus, the outer radius is the distance from the curve y=x2-2x to the axis of rotation y= 2 and the inner radius is the distance from the curve y=xcosπx4 to the axis of rotation y= 2.

This gives - outer radius = 2-x2-2x and inner radius is 2-xcosπx4

By using concept,

V=π∫022-x2-2x2-2-xcosπx42dx

By using a CAS (refer exercise 37),

V=419π2+120π-21015π

**Conclusion:**

The exact volume of the solidis

V=419π2+120π-21015π