Riemann Sums
Approximate area with rectangles.
Key idea
Approximate area with rectangles.
In depth
Approximate area by dividing an interval into n strips, forming rectangles of height f(xᵢ*) and width Δx, then adding them. As n → ∞ the sum converges to the exact area.
Worked example
f(x) = x² on [0, 1] with n = 4, right endpoints: (1/4)(1/16 + 4/16 + 9/16 + 16/16) = 30/64 ≈ 0.469. The true area is 1/3.
Formula
Real-world uses
- Estimating the area of an irregular field or the total rainfall from hourly measurements
Common mistake
Using the wrong width Δx = (b − a)/n.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Riemann Sums?
Approximate area with rectangles. It is a topic at the advanced level in Calculus.
What is the formula for Riemann Sums?
The key formula is Σ f(xᵢ*)Δx.
Where is Riemann Sums used in real life?
Real-world examples: Estimating the area of an irregular field or the total rainfall from hourly measurements.
What should I learn before Riemann Sums?
Make sure you are comfortable with Indefinite Integral Rules, the previous topic in Calculus.
What is a common mistake with Riemann Sums?
Using the wrong width Δx = (b − a)/n.
How can I practise Riemann Sums?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Riemann Sums?
Move on to Definite Integral next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.