Riemann Sums Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Riemann Sums.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A method of approximating the definite integral \int_a^b f(x)\,dx by dividing the interval [a, b] into subintervals and summing the areas of rectangles (or trapezoids) whose heights are determined by the function.

Imagine filling the area under a curve with thin rectangles. The more rectangles you use, the better the approximation. In the limit of infinitely many infinitely thin rectangles, you get the exact area—which is the definite integral.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Riemann sums are the bridge between discrete summation and continuous integration. The definite integral IS the limit of Riemann sums as n \to \infty.

Common stuck point: For an increasing function, left sums underestimate and right sums overestimate. For a decreasing function, it's the opposite. Midpoint and trapezoidal sums are generally more accurate.

Sense of Study hint: Draw the rectangles on a graph of the function to see whether your sum is an overestimate or underestimate.

Worked Examples

Example 1

easy
Approximate \int_0^2 x^2\,dx using a left Riemann sum with n = 4 equal subintervals.

Solution

  1. 1
    \Delta x = 0.5. Left endpoints: x = 0, 0.5, 1, 1.5.
  2. 2
    f(0)=0,\; f(0.5)=0.25,\; f(1)=1,\; f(1.5)=2.25.
  3. 3
    L_4 = 0.5(0+0.25+1+2.25) = 0.5 \times 3.5 = 1.75.
  4. 4
    Exact value: \frac{8}{3} \approx 2.667. The left sum underestimates because f is increasing.

Answer

L_4 = 1.75 (underestimate; exact = \frac{8}{3})
For an increasing function, left endpoints give the minimum in each subinterval, so the left Riemann sum underestimates. More subintervals improve accuracy.

Example 2

medium
Approximate \int_1^3 \frac{1}{x}\,dx using a right Riemann sum with n = 4 subintervals and classify the estimate.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Use a midpoint sum with n=2 to approximate \int_0^4 (x+1)\,dx.

Example 2

hard
Write the right Riemann sum for \int_0^1 x^3\,dx with n subintervals as a sigma expression and evaluate the limit.
← Back to Riemann Sums Formula Explained Practice Mode

Related Concepts

IntegralDefinite Integral

Background Knowledge

These ideas may be useful before you work through the harder examples.

integraldefinite integral