Lecture on Determining Volumes by Slicing

Introduction

• Use definite integrals to find volumes of three-dimensional solids.
• Methods: Slicing, Disk, and Washer.

Volume and the Slicing Method

• Volume Definition: Numerical measure of a three-dimensional solid.
• Basic Volume Formulas:
  • Rectangular Solid: ( V = lwh )
  • Sphere: ( V = \frac{4}{3}\pi r^3 )
  • Cone: ( V = \frac{1}{3}\pi r^2 h )
  • Pyramid: ( V = \frac{1}{3}A h )
• Cylinders:
  • General definition: Solid with identical cross-sections perpendicular to its axis.
  • Volume: ( V = A \cdot h )
• Slicing Method:
  • Slice solid into pieces, estimate each slice's volume, sum them up.
  • Volume approximation using Riemann sum, limit as slices approach infinity.

Strategy for Slicing Method

1. Determine shape of cross-section.
2. Find formula for cross-sectional area.
3. Integrate area formula over the interval.

Examples

Example 2.6: Volume of a Pyramid

• Pyramid with Square Base: Cross-sections are squares.
• Area Formula: ( A(x) = \left(\frac{ax}{h}\right)^2 )
• Integration:
  • ( V = \int_0^h \frac{a^2x^2}{h^2} , dx = \frac{1}{3}a^2 h )

Solids of Revolution

• Definition: Solid formed by revolving a region around a line.
• Applications: Common in mechanical parts designing.

Example 2.7: Solid of Revolution

• Function: ( f(x) = x^2 - 4x + 5 ), interval ([1, 4]).
• Cross-sections: Circles, radius ( f(x) ).
• Volume:
  • ( V = \int_1^4 (x^2 - 4x + 5)^2 , dx )

The Disk Method

• When to Use: When cross-sections are circles (disks).
• Volume Formula:
  • ( V = \pi \int_a^b [f(x)]^2 , dx )

Example 2.8: Disk Method

• Function: ( f(x) = x ), interval ([1, 4]).
• Volume: ( V = \int_1^4 x^2 , dx = \frac{15}{2} )

The Washer Method

• Use Case: Solids with cavities.
• Volume Formula:
  • ( V = \pi \int_a^b \left([f(x)]^2 - [g(x)]^2\right) , dx )

Example 2.10: Washer Method

• Functions: ( f(x) = x ) and ( g(x) = \frac{1}{x} ), interval ([1, 4]).
• Volume: ( V = \pi \int_1^4 (x^2 - \frac{1}{x^2}) , dx = \frac{81}{4} )

Exercises

1. Derive formulas using slicing and disk methods.
2. Solve problems involving rotating regions around axes.
3. Calculate volumes of solids with various cross-sectional shapes.