Calculating the Volume of a Solid of Revolution Using Cylindrical Shells

In this article, we will explore how to calculate the volume of a solid generated by revolving a region around a given line using the method of cylindrical shells. Specifically, we will use the curves x y2 - 2 and x 6 - y2 revolved around the line x 2. This method is a powerful tool in calculus, particularly for solving problems involving volumes of solids of revolution.

Problem Statement

The region bounded by the curves x y2 - 2 and x 6 - y2 is to be revolved around the line x 2. The goal is to determine the volume of the solid generated by this revolution.

Step 1: Finding Points of Intersection

The first step is to find the points of intersection of the curves. To do this, we set the equations equal to each other:

Equation:

y2 - 2 6 - y2

Rearranging the equation:

Step-by-Step Calculation:

2y2 - 8 0 Rightarrow y2 4 Rightarrow y plusmn;2

The points of intersection are at y 2 and y -2. This gives us the bounds for our integral.

Step 2: Setting Up the Volume Integral

The volume of the solid of revolution can be found using the method of cylindrical shells. The volume V is given by:

Volume Integral:

V 2pi; int;ab rh dy

Here,

- r is the distance from the line of rotation to the shell, and

- h is the height of the shell.

Height of the Shell (h):

For the given curves, the height h is the difference between the two curves:

Calculations:

h 6 - y2 - (y2 - 2) 8 - 2y2

Radius (r):

The radius r is the distance from the line x 2 to the curves x y2 - 2 and x 6 - y2:

r 2 - (y2 - 2) 4 - y2

r 2 - (6 - y2) y2 - 4

We'll use the radius for the left curve:

r 4 - y2

Step 3: Calculating the Volume

The volume integral becomes:

Integral:

V 2pi; int;-22 (4 - y2) (8 - 2y2) dy

Expanding the integrand:

4 - y2 (8 - 2y2) 32 - 8y2 - 4y2 - 2y4 32 - 12y2 - 2y4

Now we need to integrate each term:

Integration:

int;-22 32 dy 32[y] -22 32(2 - (-2)) 32 times; 4 128

int;-22 12y2 dy 12 left[ frac{y3}{3} right] -22 12 left( frac{23}{3} - frac{(-2)3}{3} right) 12 left( frac{8}{3} - frac{-8}{3} right) 12 left( frac{16}{3} right) 64

int;-22 2y4 dy 2 left[ frac{y5}{5} right] -22 2 left( frac{25}{5} - frac{(-2)5}{5} right) 2 left( frac{32}{5} - frac{-32}{5} right) 2 left( frac{64}{5} right) frac{128}{5}

Substituting these results back into the volume formula:

V 2pi; left( 128 - 64 - frac{128}{5} right) 2pi; left( 64 - frac{128}{5} right) 2pi; left( frac{320}{5} - frac{128}{5} right) 2pi; left( frac{448}{5} right) frac{896pi;}{5}

Final Answer

The volume of the solid generated by revolution is:

V boxed{frac{896pi}{5}}

This comprehensive solution demonstrates the step-by-step process of using cylindrical shells to find the volume of a solid of revolution, a fundamental technique in calculus.