Overview
This lecture covers how to use the shell method to find volumes of solids of revolution, focusing on rotation about the y-axis and x-axis, along with two example problems.
Shell Method Fundamentals
- The shell method finds volumes by integrating cylindrical shells around an axis.
- Always draw the rectangle (generating the shell) parallel to the axis of rotation.
- For rotation about the y-axis, the radius and height must be expressed in terms of x.
- The volume formula for rotation about the y-axis:
( V = 2\pi \int_a^b \text{radius}(x) \times \text{height}(x) , dx )
- If bounded by two curves ( f(x) ) (top) and ( g(x) ) (bottom), the height is ( f(x) - g(x) ).
Rotation About the X-Axis
- Draw the shell-generating rectangle parallel to the x-axis.
- For rotation about the x-axis, use y as the integration variable.
- The volume formula for rotation about the x-axis:
( V = 2\pi \int_c^d \text{radius}(y) \times \text{height}(y) , dy )
- If bounded by two functions ( f(y) ) (right) and ( g(y) ) (left), the height is ( f(y) - g(y) ).
Example 1: ( y = \sqrt{x} ), ( y = 0 ), ( x = 4 ), Rotated About Y-Axis
- Visualize and draw the region and a rectangle parallel to the y-axis.
- Here, radius ( r = x ), height ( h = y = \sqrt{x} ).
- Set up the integral: ( V = 2\pi \int_0^4 x \sqrt{x} , dx ).
- Combine exponents: ( x \sqrt{x} = x^{3/2} ).
- Integrate: Antiderivative is ( (2/5)x^{5/2} ).
- Evaluate from 0 to 4: ( 2\pi[(2/5) \cdot 32 - 0] = 128\pi / 5 ).
Example 2: ( y = x - x^3 ), ( y = 0 ), ( x = 0 ) to ( x = 1 ), Rotated About Y-Axis
- Find x-intercepts: 0, -1, 1; focus on ( 0 \leq x \leq 1 ).
- Radius ( r = x ), height ( h = y = x - x^3 ).
- Set up the integral: ( V = 2\pi \int_0^1 x(x - x^3) , dx = 2\pi \int_0^1 (x^2 - x^4) , dx ).
- Integrate: Antiderivative is ( x^3/3 - x^5/5 ).
- Evaluate from 0 to 1: ( 2\pi[(1/3 - 1/5)] = 4\pi/15 ).
Key Terms & Definitions
- Shell Method — A technique to find the volume of solids by integrating cylindrical shells.
- Radius (of shell) — Distance from the axis of rotation to the rectangle generating the shell.
- Height (of shell) — Difference between the top and bottom functions (or right and left, if rotating about x-axis).
Action Items / Next Steps
- Practice setting up and evaluating shell method integrals for various axis rotations and regions.