Sample Answer

To find the volume of the solid generated by revolving the area bounded by (0,0), (4,8), (4,0) about the x-axis, we can use the disk method.

Step 1: Set up the integral:

The disk method formula for finding the volume of a solid of revolution is:

V = π ∫[a,b] (R(x))^2 dx

where:

• V is the volume of the solid
• R(x) is the radius of the disk at a given x-value
• a and b are the limits of integration

In this case, R(x) = y = x^(3/2). The limits of integration are a = 0 and b = 4.

Step 2: Evaluate the integral:

V = π ∫[0,4] (x^(3/2))^2 dx
  = π ∫[0,4] x^3 dx
  = π [x^4/4] |[0,4]
  = π (4^4/4 - 0^4/4)
  = 64π