generated solid

Question

The region in the first quadrant bounded by the x-axis, the line x = π, and the curve y = cos(cos(x)) is rotated about the x-axis. What is the volume of the generated solid? 

1.921

3.782

6.040

8.130

Answer 1

This is a solid of extrusion problem. Or at least, this is what an engineer would call it. The solid has a cross-section that is a circular disk - what you get when you revolve the defined function around the x-axis. You get an incremental volume of the solid, which is the area of the cross section multiplied by an increment of thickness, along the x-axis. Then you integrate from left to right.

 

The cross-section has a simple shape: a circle. The area is π times radius squared, and the radius is your function: r = cos (cos (x)). so

 

A = π [cos (cos (x))]^2

 

Once you have that, you integrate from left (x=0) to right (x=π) to find volume:

 

V = ∫ A dx

 

V = (1/2) π^2 (1 + J_0 (2)) where J_0 (2) is a Bessel function, and there's probably a table look-up for that in your textbook. Me, I used Wolfram Alpha to figure it out. LOL

 

V = 6.03966

There you go. :-)