Calculus

  1. (10 points) The equation of the red curve is y2 = x3. Find the volume generated when area bounded by (0,0), (4,8), (4,0) is revolved about the x-axis. (similar to 5.1 #7-12)

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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:

 

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.

 

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