BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon102 Learners

Last updated on August 5th, 2025

Math Whiteboard Illustration

Math Formula for Riemann Sum

Professor Greenline Explaining Math Concepts

In calculus, the Riemann sum is a method for approximating the integral of a function over an interval. It is a crucial concept for understanding the area under curves and for the development of the definite integral. In this topic, we will learn the Riemann sum formula and its application.

Math Formula for Riemann Sum for Australian Students
Professor Greenline from BrightChamps

List of Math Formulas for Riemann Sum

The Riemann sum is a way to approximate the area under a curve by dividing it into small segments. Let’s learn the formula to calculate the Riemann sum.

Professor Greenline from BrightChamps

Math Formula for Riemann Sum

The Riemann sum is an approximation of the integral of a function over an interval [a, b].

 

It is calculated using the formula: [ S = sum_{i=1}^{n} f(x_i^*) \Delta x ] where ( x_i^* ) is a sample point in the i-th subinterval, (Delta x = frac{b-a}{n}) is the width of each subinterval, and ( n ) is the number of subintervals.

Professor Greenline from BrightChamps

Types of Riemann Sums

There are different types of Riemann sums based on the choice of sample points:

 

- Left Riemann Sum: Uses the left endpoint of each subinterval.

 

- Right Riemann Sum: Uses the right endpoint of each subinterval.

 

- Midpoint Riemann Sum: Uses the midpoint of each subinterval.

Professor Greenline from BrightChamps

Importance of Riemann Sum Formula

In math and real world applications, the Riemann sum formula is essential for approximating integrals and understanding areas under curves. Here are some important aspects of the Riemann sum:

 

- It helps in the numerical approximation of integrals where an analytical solution is difficult.

 

- By learning this formula, students can grasp concepts like definite integrals, areas under curves, and integral calculus.

 

- Engineers and scientists use Riemann sums to model and solve real-world problems involving continuous data.

Professor Greenline from BrightChamps

Tips and Tricks to Understand Riemann Sum Formula

Students may find the concept of Riemann sums tricky and confusing. Here are some tips and tricks to master the Riemann sum formula:

 

- Visualize the process by drawing the graph and dividing it into rectangles.

 

- Start with simple functions and calculate the Riemann sum to build confidence.

 

- Use software tools or calculators to compute Riemann sums for complex functions.

Professor Greenline from BrightChamps

Real-Life Applications of Riemann Sum Formula

In real life, Riemann sums play a major role in approximating areas and solving practical problems. Here are some applications of the Riemann sum formula:

 

- In physics, to calculate the work done by a variable force, we use Riemann sums.

 

- In economics, to estimate consumer surplus, Riemann sums are applied.

 

- In environmental science, to approximate the total pollutant discharge over time, Riemann sums are useful.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them While Using Riemann Sum Formula

Students make errors when calculating Riemann sums. Here are some mistakes and the ways to avoid them, to master them.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Choosing Incorrect Sample Points

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes choose incorrect sample points, which leads to an inaccurate Riemann sum. To avoid this error, consistently use the correct left, right, or midpoint sample point for your calculations.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Miscalculating \(\Delta x\)

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

When calculating \(\Delta x\), students may make errors in determining the width of each subinterval. To avoid these errors, ensure that \(\Delta x = \frac{b-a}{n}\) is calculated correctly.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrectly Summing the Terms

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students may sum the terms incorrectly or miss some terms. To avoid this, carefully sum all terms and double-check your work.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing Types of Riemann Sums

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students usually confuse the different types of Riemann sums, leading to errors. To avoid this confusion, understand the difference between left, right, and midpoint Riemann sums and when to use each.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Skipping Visualization

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes skip visualizing the problem, which can lead to misunderstanding. To avoid this, sketch the function and the subintervals to better understand the problem.

arrow-right
Max from BrightChamps Saying "Hey"
Hey!

Examples of Problems Using Riemann Sum Formula

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Approximate the integral of \( f(x) = x^2 \) on the interval [1, 3] using a left Riemann sum with 4 subintervals.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The approximate integral is 5.5

Explanation

Divide the interval [1, 3] into 4 subintervals: \(\Delta x = \frac{3-1}{4} = 0.5\).

Using left endpoints: \[ S = f(1) \cdot 0.5 + f(1.5) \cdot 0.5 + f(2) \cdot 0.5 + f(2.5) \cdot 0.5 = 1 \cdot 0.5 + 2.25 \cdot 0.5 + 4 \cdot 0.5 + 6.25 \cdot 0.5 = 5.5 \]

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 2

Estimate the area under \( f(x) = 2x + 1 \) from x = 0 to x = 2 using a right Riemann sum with 2 subintervals.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The estimated area is 8

Explanation

Divide the interval [0, 2] into 2 subintervals: \(\Delta x = \frac{2-0}{2} = 1\).

Using right endpoints: \[ S = f(1) \cdot 1 + f(2) \cdot 1 = 3 \cdot 1 + 5 \cdot 1 = 8 \]

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 3

Find the midpoint Riemann sum for \( f(x) = x \) over [0, 4] with 4 subintervals.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The midpoint Riemann sum is 8

Explanation

Divide the interval [0, 4] into 4 subintervals: \(\Delta x = \frac{4-0}{4} = 1\).

Midpoints: 0.5, 1.5, 2.5, 3.5 \[ S = f(0.5) \cdot 1 + f(1.5) \cdot 1 + f(2.5) \cdot 1 + f(3.5) \cdot 1 = 0.5 \cdot 1 + 1.5 \cdot 1 + 2.5 \cdot 1 + 3.5 \cdot 1 = 8 \]

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 4

Calculate the left Riemann sum of \( f(x) = 3x \) over [2, 5] with 3 subintervals.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The left Riemann sum is 31.5

Explanation

Divide the interval [2, 5] into 3 subintervals: \(\Delta x = \frac{5-2}{3} = 1\).

Using left endpoints: \[ S = f(2) \cdot 1 + f(3) \cdot 1 + f(4) \cdot 1 = 6 \cdot 1 + 9 \cdot 1 + 12 \cdot 1 = 31.5 \]

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 5

Approximate the integral of \( f(x) = x^3 \) from 0 to 1 using a midpoint Riemann sum with 2 subintervals.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The approximate integral is 0.28125

Explanation

Divide the interval [0, 1] into 2 subintervals: \(\Delta x = \frac{1-0}{2} = 0.5\).

Midpoints: 0.25, 0.75 \[ S = f(0.25) \cdot 0.5 + f(0.75) \cdot 0.5 = 0.015625 \cdot 0.5 + 0.421875 \cdot 0.5 = 0.28125 \]

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Ray Thinking Deeply About Math Problems

FAQs on Riemann Sum Formula

1.What is the Riemann sum formula?

The formula for the Riemann sum is: \[ S = \sum_{i=1}^{n} f(x_i^*) \Delta x \] where \( x_i^* \) is a point in the i-th subinterval, and \(\Delta x = \frac{b-a}{n}\).

Math FAQ Answers Dropdown Arrow

2.What are the types of Riemann sums?

The types of Riemann sums include the left Riemann sum, right Riemann sum, and midpoint Riemann sum, differentiated by the sample points used.

Math FAQ Answers Dropdown Arrow

3.How do you calculate the left Riemann sum?

To calculate the left Riemann sum, use the left endpoint of each subinterval to evaluate the function and sum the results multiplied by \(\Delta x\).

Math FAQ Answers Dropdown Arrow

4.What is the right Riemann sum?

The right Riemann sum uses the right endpoint of each subinterval to evaluate the function and is calculated by summing these values multiplied by \(\Delta x\).

Math FAQ Answers Dropdown Arrow

5.How is the midpoint Riemann sum different?

The midpoint Riemann sum uses the midpoint of each subinterval to evaluate the function, providing a potentially more accurate approximation of the integral.

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Glossary for Riemann Sum Formula

  • Riemann Sum: An approximation of the integral of a function over an interval by dividing it into small subintervals and summing the areas of rectangles.

     
  • Subinterval: A smaller division of the overall interval used in Riemann sums.

     
  • Left Riemann Sum: Uses the left endpoints of subintervals for evaluation.

     
  • Right Riemann Sum: Uses the right endpoints of subintervals for evaluation.

     
  • Midpoint Riemann Sum: Uses the midpoints of subintervals for evaluation, often providing a more accurate approximation.
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
UAE - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom