4.2.1: Resources and Key Concepts

Prerequisite Topics

The following is a list of prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Sequences and Series
  • Summations and Their Properties: Sigma notation (\(\Sigma\)) is introduced and used extensively. Properties of summations are listed and applied.

Videos

• Sigma notation
  • Sigma Notation and Its Properties
  • Using Sigma Notation
• Properties of summation
  • Review of Summation Properties Required for Integral Calculus
• Approximating areas
  • Approximating Areas Under Curves
• Riemann sum
  • Riemann Sums
  • Computing Areas “Under” Curves Using Riemann Sums
  • Computing Distances Using Riemann Sums

Definitions

• Sigma Notation (Summation Notation): A compact way to express long sums of values using the Greek capital letter \(\Sigma\). In \( \displaystyle \sum_{i=1}^{n} a_i\), \(a_i\) describes the terms to be added, and \(i\) is the index (a dummy variable).
• Partition of an Interval: A set of points \(P = \{x_i\}\) for \(i=0, 1, 2, \dots, n\) with \(a = x_0 < x_1 < x_2 < \dots < x_n = b\), which divides the interval \([a,b]\) into \( n \) subintervals of the form \([x_{i-1}, x_i]\).
• Regular Partition (Uniform Partition): A partition where all subintervals have the same width, \(\Delta x = \frac{b-a}{n}\).
• Left-Endpoint Approximation (\(L_n\)): An approximation of the area under a curve using rectangles where the height of each rectangle on the subinterval \([x_{i-1}, x_i]\) is determined by the function value at the left endpoint, \(f(x_{i-1})\). \(A \approx L_n = \displaystyle \sum_{i=1}^{n} f(x_{i-1})\Delta x\).
• Right-Endpoint Approximation (\(R_n\)): An approximation of the area under a curve using rectangles where the height of each rectangle on the subinterval \([x_{i-1}, x_i]\) is determined by the function value at the right endpoint, \(f(x_i)\). \(A \approx R_n = \displaystyle \sum_{i=1}^{n} f(x_i)\Delta x\).
• Riemann Sum: Let \(f(x)\) be defined on a closed interval \([a,b]\) and let \(P\) be any partition of \([a,b]\). Let \(\Delta x_i\) be the width of each subinterval \([x_{i-1}, x_i]\) and for each \(i\), let \(x_i^*\) be any point in \([x_{i-1}, x_i]\). A Riemann sum is defined for \(f(x)\) as \( \displaystyle \sum_{i=1}^{n} f(x_i^*)\Delta x_i\).
• Upper Sum: A Riemann sum where \(f(x_i^*)\) is the maximum function value on the interval \([x_{i-1}, x_i]\).
• Lower Sum: A Riemann sum where \(f(x_i^*)\) is the minimum function value on the interval \([x_{i-1}, x_i]\).
• Method of Exhaustion: An ancient process of approximating areas of irregular regions by filling them with shapes of known area, and then summing these areas.

Theorems

• Theorem: Properties of Sigma Notation:
  1. \( \displaystyle \sum_{i=1}^{n} c = nc\)
  2. \( \displaystyle \sum_{i=1}^{n} ca_i = c \displaystyle \sum_{i=1}^{n} a_i\)
  3. \( \displaystyle \sum_{i=1}^{n} (a_i+b_i) = \displaystyle \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i\)
  4. \( \displaystyle \sum_{i=1}^{n} (a_i-b_i) = \displaystyle \sum_{i=1}^{n} a_i - \displaystyle \sum_{i=1}^{n} b_i\)
  5. \( \displaystyle \sum_{i=1}^{n} a_i = \displaystyle \sum_{i=1}^{m} a_i + \displaystyle \sum_{i=m+1}^{n} a_i\) (for \(1 \le m < n\))
• Theorem: Sums and Powers of Integers:
  1. Sum of integers: \( \displaystyle \sum_{i=1}^{n} i = \frac{n(n+1)}{2}\)
  2. Sum of consecutive integers squared: \( \displaystyle \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\)
  3. Sum of consecutive integers cubed: \( \displaystyle \sum_{i=1}^{n} i^3 = \left[\frac{n(n+1)}{2}\right]^2\)
• Theorem: Left-Endpoint Approximation (Formula stated under Definitions).
• Theorem: Right-Endpoint Approximation (Formula stated under Definitions).
• Area Under the Curve: Let \(f(x)\) be a continuous, nonnegative function on an interval \([a,b]\), and let \( \displaystyle \sum_{i=1}^{n} f(x_i^*)\Delta x\) be a Riemann sum for \(f(x)\) with a regular partition \(P\). Then, the area under the curve \(y=f(x)\) on \([a,b]\) is given by \(A = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\Delta x\).
• (Uniqueness of Limit of Riemann Sums): Although not formally stated as a theorem with proof in this section, the text mentions that if \(f(x)\) is continuous on the closed interval \([a,b]\), then \(\displaystyle \lim_{n \to \infty} \displaystyle \sum_{i=1}^{n} f(x_i^*)\Delta x\) exists and is unique (does not depend on the choice of \(x_i^*\)).

Common Mistakes

• Incorrectly Using Sigma Notation Properties: For example, assuming \( \sum (a_i b_i) = (\sum a_i)(\sum b_i)\).
• Confusing Left and Right Endpoints: Using \(f(x_i)\) for a left-endpoint sum or \(f(x_{i-1})\) for a right-endpoint sum.
• Not Using Enough Rectangles for Good Approximation: Realizing that approximations with a small number of rectangles (\(n\)) can be quite inaccurate.
• Assuming Riemann Sum is Exact Area: A Riemann sum with a finite \(n\) is an approximation, not the exact area (unless the function is constant or the rectangles perfectly fit). The exact area is the limit as \(n \to \infty\).
• Treating the Index of Summation as a Factor: The index (e.g., \(i\) in \( \displaystyle \sum a_i\)) is a dummy variable and does not factor into the sum itself beyond indicating which terms are added.