Math Problem Statement

Let R be the region bounded by y=x2 , x=0 , and y=4 . Set up the integral that represents the volume of the solid formed when R is revolved about the line x=4 using the Shell Method.

Solution

To set up the integral for the volume of the solid formed when the region RR is revolved about the line x=4x = 4 using the Shell Method, we proceed as follows:

Step 1: Analyze the given region RR

The region RR is bounded by:

  • y=x2y = x^2 (a parabola),
  • x=0x = 0 (the vertical line),
  • y=4y = 4 (a horizontal line).

This means that the region starts from x=0x = 0 and extends to x=2x = 2 (since y=4y = 4 intersects y=x2y = x^2 at x=2x = 2).

Step 2: Shell Method formula

The formula for the volume of a solid using the Shell Method when revolving around a vertical line x=ax = a is:

V=2πx1x2(radius of shell)×(height of shell)dxV = 2\pi \int_{x_1}^{x_2} \text{(radius of shell)} \times \text{(height of shell)} \, dx

In this case, we are revolving about x=4x = 4.

  • Radius of each shell: The distance from a point xx to the line x=4x = 4. This is given by 4x|4 - x|. Since the region is between x=0x = 0 and x=2x = 2, the radius simplifies to 4x4 - x.

  • Height of each shell: The height is the difference between the top and bottom of the region at a given xx-value. The top of the region is given by y=4y = 4, and the bottom of the region is given by y=x2y = x^2. So, the height of the shell is 4x24 - x^2.

Step 3: Set up the integral

Now, applying the Shell Method formula:

V=2π02(4x)(4x2)dxV = 2\pi \int_0^2 (4 - x) (4 - x^2) \, dx

Final Answer:

V=2π02(4x)(4x2)dxV = 2\pi \int_0^2 (4 - x) (4 - x^2) \, dx

Would you like to solve this integral step by step? Or do you have any questions about the setup?

Related Questions:

  1. How do you evaluate this integral?
  2. How does the Disk Method compare to the Shell Method for this problem?
  3. How do you apply the Shell Method when revolving around a horizontal line?
  4. What would the volume be if the region was revolved around the

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Math Problem Analysis

Mathematical Concepts

Calculus
Volume of Solids
Shell Method

Formulas

V = 2π ∫(radius of shell)(height of shell) dx

Theorems

Shell Method

Suitable Grade Level

Grades 11-12