Riemann Sums Explained: Left, Right, Midpoint and Trapezoid

A Riemann sum estimates the area under a curve by splitting it into thin strips and adding up rectangles. The height of each rectangle comes from the function at the left end, right end or middle of its strip. As the strips get thinner, the sum approaches the exact integral.

Setting up

To approximate $\int_a^b f(x)\,dx$ with $n$ strips, each strip has width

Worked example: $\int_0^2 x^2\,dx$

The exact area is $\frac{8}{3} \approx 2.667$.

Left sum, $n = 4$: $\Delta x = 0.5$, heights at $0, 0.5, 1, 1.5$:

This is too small, because $x^2$ is increasing, so left heights are the lowest in each strip.

Midpoint sum, $n = 8$: $\Delta x = 0.25$, heights at $0.125, 0.375, \ldots, 1.875$. The total is $2.65625$, already very close.

Comparing the methods

From sums to integrals

The definite integral is defined as the limit of Riemann sums:

That is why the integral symbol is a stretched S, for "sum".

Common mistakes

• Using $n + 1$ rectangles instead of $n$.
• Mixing up which end to use for left and right sums.
• Forgetting to multiply by $\Delta x$.

Practice questions

1. Find the right sum for $x^2$ on $[0, 2]$ with $n = 4$.
2. Find the trapezoid estimate for $x$ on $[0, 4]$ with $n = 2$.
3. Is a left sum of $\frac{1}{x}$ on $[1, 3]$ an over- or underestimate?

Answers: 1) $3.75$ 2) $8$ (exact) 3) overestimate

Draw the rectangles yourself in the Riemann sum calculator, then compare with the exact value from the integral calculator. See also integration for beginners.

Frequently asked questions

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