Riemann Sums and the Definition of the Definite Integral Notes

Course Information and Learning Objectives

• Course: MATH 142 - CALCULUS II
• Term: SEMESTER III
• Topic: RIEMANN SUMS AND THE DEFINITION OF THE DEFINITE INTEGRAL
• Session: TUTORIAL 1
• Target Audience: ALL COHORTS
• Date: 22 MAY 2026

Primary Objectives

• Develop an understanding of the application and use of summation notation ($\Sigma$).
• Utilize rectangles to approximate the area under a graph through the method of Riemann summation.
• Apply the concept of finding the area under a graph to resolve practical real-world problems.

The Process of Riemann Summation

To approximate the area under a curve using Riemann summation, the following procedural steps must be followed:

1. Graphing: Draw the graph of the function $f(x)$.
2. Subdivision: Subdivide the specified interval $[a, b]$ into $n$ subintervals of equal width. The width (horizontal length) of each rectangle is calculated using the formula:     $\Delta x = \frac{b-a}{n}$
3. Construction: Construct rectangles above the subintervals. For a left-endpoint approximation, ensure that the top left corner of each rectangle touches the graph of the function.
4. Area Calculation: Determine the area of each individual rectangle. Since the width is constant ($\Delta x$) and the height is determined by the function value at a specific point ($x_i$), the area of one rectangle is $f(x_i) \times \Delta x$.
5. Summation: Sum these individual areas to achieve an approximation for the total area under the curve over the interval $[a, b]$.

Exercises in Summation and Sigma Notation

Q1. Conversion to Summation Notation

• Write the following expressions using sigma ($\Sigma$) notation:
  • i. $3 + 6 + 9 + 12 + 15 + 18$
  • ii. $5 + 10 + 15 + 20 + 25 + 30 + 35$
  • iii. $f(x_1) + f(x_2) + f(x_3) + f(x_4)$
  • iv. $g(x_1) + g(x_2) + g(x_3) + g(x_4) + g(x_5)$
  • v. $G(x_1) + G(x_2) + \dots + G(x_{15})$
  • vi. $F(x_1) + F(x_2) + \dots + F(x_{17})$

Q1b. Series Analysis in Sigma Notation

• Determine the sigma notation for the following series:
  • i. $\frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \dots$
  • ii. $\frac{1}{4} - \frac{1}{9} + \frac{1}{16} - \dots$
  • iii. $5 + 6 + 7 + 8 + 9$

Q1c. Expanding Sums

• Expand the following sigma expressions:
  • i. $\sum_{i=1}^{4} 3i$
  • ii. $\sum_{j=1}^{3} n^j$
  • iii. $\sum_{j=1}^{3} \frac{3}{n}$

Q1d. Finding Closed Form Values

• Find the numerical or simplified algebraic values for:
  • i. $\sum_{i=1}^{n} i$
  • ii. $\sum_{j=1}^{1,000} (j+1)$
  • iii. $\sum_{k=1}^{n} (6k^2 - 3k + 4)$

Q2. Expression Without Summation Notation

• Fully expand the following summations:
  • i. $\sum_{i=1}^{4} i^2$
  • ii. $\sum_{i=0}^{5} (-2)^i$
  • iii. $\sum_{i=1}^{5} f(x_i)$
  • iv. $\sum_{i=1}^{4} g(x_i)$

Graphical Area Approximations

Q3. Functions of the Form $f(x) = \frac{1}{x^2}$

• Part a: Approximate the area under the graph of $f(x) = \frac{1}{x^2}$ over the interval $[1, 7]$.
  • Process: Compute the area of each rectangle to four decimal places and calculate their sum.
• Part b: Approximate the area under the same graph ($f(x) = \frac{1}{x^2}$) over the interval $[1, 7]$ using a different subdivision method (implied by the request to compare).
  • Process: Compute to four decimal places and compare the resulting value to the answer obtained in Part a.

Q4. Functions of the Form $f(x) = x^2 + 1$

• Part a: Approximate the area under the graph of $f(x) = x^2 + 1$ over the interval $[0, 5]$ by computing individual rectangle areas to four decimal places and summing them.
• Part b: Provide another approximation for the area under $f(x) = x^2 + 1$ over the interval $[0, 5]$ and compare this result to the value found in Part a.

Real-World Applications of Riemann Sums

Q5. Raggs, Ltd. - Manufacturing Costs

• Scenario: The company has determined that its marginal cost for the $x^{\text{th}}$ jacket produced is $C'(x) = 0.0003x^2 - 0.2x + 50$.
• Task: Approximate the total cost of producing $400$ jackets.
• Parameters:
  • Summation: $\sum_{i=1}^{4} C'(x_i) \Delta x_i$
  • Width: $\Delta x = 100$

Q6. Holcomb Hill Fitness - Profit Analysis

• Scenario: The marginal profit $P'(x)$ (in cents) is given by $P'(x) = -0.0006x^3 + 0.28x^2 + 55.6x$, defined for $x \le 500$, where $x$ is the number of enrolled members.
• Task: Approximate the total profit when $300$ members are enrolled.
• Parameters:
  • Summation: $\sum_{i=1}^{6} P'(x_i) \Delta x$
  • Width: $\Delta x = 50$

Q7. Ship Shape Woodworkers - Custom Molding

• Scenario: Marginal cost for producing custom molding (in cents) is $C'(x) = -0.00002x^2 - 0.04x + 45$ for $x \le 800$.
• Task: Approximate the total cost of manufacturing $800\text{ ft}$ of molding.
• Parameters:
  • Intervals: $5$ subintervals.
  • Domain: $[0, 800]$.
  • Method: Use the left endpoint of each subinterval.

Q8. Soulful Scents - Fragrance Production

• Scenario: The marginal cost of producing $x$ ounces of fragrance (in dollars) is $C'(x) = 0.005x^2 - 0.1x + 30$ for $x \le 125$.
• Task: Approximate the total cost of producing $100\text{ oz}$ of fragrance.
• Parameters:
  • Intervals: $5$ subintervals.
  • Domain: $[0, 100]$.
  • Method: Use the left endpoint of each subinterval.

Q9. Shelly’s Roadside Fruit - Orange Juice Production

• Scenario: The marginal cost of producing $x$ pints of orange juice (in dollars) is $C'(x) = 0.000008x^2 - 0.004x + 2$ for $x \le 350$.
• Task: Approximate the total cost of producing $270\text{ pt}$ of juice.
• Parameters:
  • Intervals: $3$ subintervals.
  • Domain: $[0, 270]$.
  • Method: Use the left endpoint of each subinterval.

Q10. Mangianello Paving, Inc. - Road Construction

• Scenario: The marginal cost of paving a road surface with asphalt (in dollars) is $C'(x) = \frac{1}{6}x^2 - 20x + 1800$ for $x \le 80$, where $x$ is measured in hundreds of feet.
• Task: Approximate the total cost of paving $4000\text{ ft}$ of road surface.
• Parameters:
  • Distance Mapping: $4000\text{ ft} = 40$ units of hundreds of feet.
  • Intervals: $4$ subintervals.
  • Domain: $[0, 40]$.
  • Method: Use the left endpoint of each subinterval.