Riemann Sums Study Guide

1. Basic Idea

• Riemann Sum: Approximates an integral by summing the areas of vertical rectangles.
• Formula: [ \int_a^b f(x) , dx \approx f(x_1)\Delta x + f(x_2)\Delta x + \ldots + f(x_n)\Delta x ]
• ( \Delta x = \frac{b-a}{n} ): Width of each rectangle
• Common Rules for choosing x-values:
  1. Right endpoint
  2. Left endpoint
  3. Midpoint
• Example: Interval ([1, 3]) with ( n = 4 )
  • ( \Delta x = \frac{3-1}{4} = 0.5 )
  • Right endpoint: ( f(1.5)0.5 + f(2)0.5 + f(2.5)0.5 + f(3)0.5 )
  • Left endpoint: ( f(1)0.5 + f(1.5)0.5 + f(2)0.5 + f(2.5)0.5 )
  • Midpoint: ( f(1.25)0.5 + f(1.75)0.5 + f(2.25)0.5 + f(2.75)0.5 )
• Estimations:
  • If ( f(x) ) is increasing, left endpoint underestimates and right endpoint overestimates.
  • If ( f(x) ) is decreasing, left endpoint overestimates and right endpoint underestimates.

2. Summations

• Summation Notation: Describes large sums by a formula for each term.
  • Example: ( \sum_{k=1}^{10} k^2 = 1^2 + 2^2 + 3^2 + \ldots + 10^2 )
• Riemann Sum Summations:
  • Right endpoint: ( \int_a^b f(x) , dx \approx \sum_{k=1}^{n} f(a + \frac{b-a}{n} k) \frac{b-a}{n} )
  • Left endpoint: ( \int_a^b f(x) , dx \approx \sum_{k=1}^{n} f(a + \frac{b-a}{n} (k-1)) \frac{b-a}{n} )
  • Midpoint: ( \int_a^b f(x) , dx \approx \sum_{k=1}^{n} f(a + \frac{b-a}{n} (k - 0.5)) \frac{b-a}{n} )
• Limit Definition of Integral: As rectangles increase to infinity.
  • ( \int_a^b f(x) , dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(a + \frac{b-a}{n} k) \frac{b-a}{n} )

3. Exercises

1. Estimate ( \int_0^{10} f(x) , dx ) with five rectangles and left endpoints.
2. Racecar speed graph:
   • Estimate distance with five rectangles and right endpoints.
   • Determine if overestimate/underestimate.
3. Approximate ( \int_0^2 x^2 , dx ) with four rectangles, right endpoints.
4. Approximate ( \int_0^{\pi} \sin 2x , dx ) with three rectangles, midpoints.
5. Approximate ( \int_1^5 x , dx ) with 100 rectangles, right endpoints. Calculate first and last rectangle areas and determine estimate bias.
6. Function table:
   • Estimate ( \int_0^{0.5} f(x) , dx ) with right endpoints.
   • Determine estimate bias.
7. Write integrals from given Riemann sums.
8. Evaluate summation expressions.
9. Write summations for given expressions.
10. Use limit definition to express integrals as limits.

Answers

• Solutions to exercises are provided for self-verification.