Calculating Volume by Rotation: Disc and Washer Methods

Overview

• Objective: Learn to calculate the volume of a region by rotating it around the x-axis or y-axis.
• Methods: Disc Method and Washer Method.

Disc Method

• Used to calculate the volume of a solid of revolution.
• Involves slicing the solid into thin discs.
• Volume of a Cylinder: ( V = \pi r^2 h )
  • ( r ): Radius of the disc.
  • ( h ): Height (or thickness) of the disc, represented as ( \Delta x ) or ( dx ).
• Total Volume: Integrate the volume of discs from ( a ) to ( b ).
• Formula: ( V = \pi \int_{a}^{b} [r(x)]^2 , dx ) when revolving about the x-axis._

Key Points

• Cross-sectional area is circular: ( A = \pi r^2 ).
• X-axis Rotation: Radius is the function ( r(x) ).
• Y-axis Rotation: Adjust to be in terms of ( y ); use limits ( c ) to ( d ).
  • Formula: ( V = \pi \int_{c}^{d} [r(y)]^2 , dy )._

Example 1: Rotation about the X-axis

• Function: ( y = \sqrt{x} )
• Bounds: ( x = 0 ) to ( x = 4 )
• Process:
  1. Find ( r(x) = \sqrt{x} ).
  2. Set up the integral: ( V = \pi \int_{0}^{4} (\sqrt{x})^2 , dx )
  3. Simplify: ( x ).
  4. Integrate: ( \frac{x^2}{2} ) from 0 to 4.
  5. Calculate: ( 8\pi )._

Example 2: Rotation about the X-axis

• Function: ( y = \frac{1}{x} )
• Bounds: ( x = 1 ) to ( x = 3 )
• Process:
  1. ( r(x) = \frac{1}{x} ).
  2. Integral: ( V = \pi \int_{1}^{3} \left(\frac{1}{x}\right)^2 , dx )
  3. Express ( \frac{1}{x^2} ) as ( x^{-2} ).
  4. Integrate: (-\frac{1}{x} ) from 1 to 3.
  5. Calculate: ( \frac{2\pi}{3} )._

Example 3: Rotation about the Y-axis

• Function: ( y = x^2 )
• Bounds: ( y = 0 ) to ( y = 4 )
• Process:
  1. Convert: ( x = \sqrt{y} ).
  2. ( r(y) = \sqrt{y} ).
  3. Integral: ( V = \pi \int_{0}^{4} y , dy )
  4. Integrate: ( \frac{y^2}{2} ) from 0 to 4.
  5. Calculate: ( 8\pi )._

Example 4: Rotation about the Y-axis

• Function: ( y = x^{2/3} )
• Bounds: ( y = 0 ) to ( y = 1 )
• Process:
  1. Convert: ( x = y^{3/2} ).
  2. ( r(y) = y^{3/2} ).
  3. Integral: ( V = \pi \int_{0}^{1} y^3 , dy )
  4. Integrate: ( \frac{y^4}{4} ) from 0 to 1.
  5. Calculate: ( \frac{\pi}{4} )._

Conclusion

• Disc Method: Effective for calculating volumes of solids of revolution around the x or y-axis.
• Integral setup and conversion between variables are crucial steps in solving these problems.