To determine

To find:

The exact volume of the solid obtained by rotating the region bounded by the given curves about the line y= -1 by using a computer algebra system.

Answer

V= 11π28

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=sin2⁡x, y=0, 0≤x≤π; about  y= -1

3) Calculation:

Region bounded by the given curves is shown below:

From the above graph,

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

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=sin2⁡x to the axis of rotation y= -1 and the inner radius is the distance from the line y=0 to the axis of rotation y= -1.

This gives inner radius = 0--1=1 and outer radius is sin2⁡x-(-1)=sin2⁡x+1

By using the concept

=π∫0π1+sin2⁡x2-12dx

By using the command in Mathematica,

Integrate[Pi*(((1+(Sin[x])^2)^2)-1),{x,0,Pi}]

V=11π28

Therefore,

The exact volume of the solid is,

V= 11π28

Conclusion:

The exact volume of the solid is,

V= 11π28