Lecture 4 Arc Length Surface Area Partial Fractions
- Slides: 33
Lecture 4: Arc Length, Surface Area, Partial Fractions, Trigonometric Integrals, Trigonometric Substitutions, and Completing the Square
Part I: Arc Length and Surface Area
Objectives • Know how to find arc lengths and surface area Corresponding sections in Simmons: 7. 5, 7. 6
Finding Arc Lengths •
Arc Length of a Circle •
Finding Surface Area •
Surface Area of a Sphere •
Part II: Partial Fractions
Objectives • Know how to integrate rational functions (fractions whose numerators and denominators are polynomials) by expressing them as a sum of terms whose denominator is as simple as possible. Corresponding section in Simmons: 10. 6
Algebra Review: Polynomial Division • Example:
The Remainder Term •
Integrating Proper Rational Functions •
Splitting Proper Rational Functions •
Trick for finding Numerators •
Quadratics in the Denominator •
Higher Powers in the Denominator •
Part III: Trigonometric Integrals
Objectives •
• Odd powers of sinx or cosx
Integral of secx •
Even powers of sinx and cosx •
Even powers of sinx and cosx •
Even powers of sinx and cosx •
Trigonometric Substitutions
Objectives •
Key idea • The crucial idea for trigonometric substitutions is to make a substitution that makes the square root work out nicely.
Objectives •
Completing the square •
Example: •