Calculus

Full Answer Section

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

Therefore, the volume of the solid is 64π cubic units.

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: