Calculus I · Unit 3A · lesson

Left, Right, and Midpoint Riemann Sums

Concept

Learning objectives

Construct common Riemann sums, calculate them from formulas or tables, and reason about overestimates and underestimates.

Left, Right, and Midpoint Riemann Sums

Explanation

Three reasonable approximations, three different biases

Left, right, and midpoint sums use the same partition but choose different representative heights. On an increasing function, left rectangles tend to lie below the graph and right rectangles tend to lie above it; for a decreasing function, the pattern reverses. Midpoint rectangles often balance some of that error because each height is sampled from the center rather than from one edge.

The point is not that one rule is always magically correct. The shape of the function and the fineness of the partition control the quality of the estimate. Before calculating, predict whether a rule should overestimate or underestimate. After calculating, compare the result with the graph and units. Numerical integration is most useful when it combines computation with a reasoned expectation about error.

On each subinterval, choose a sample point $x_i^*$ . The rectangle contribution is

f(x_i^*)\Delta x.

Adding gives

\sum_{i=1}^n f(x_i^*)\Delta x.

Left sums use left endpoints, right sums use right endpoints, and midpoint sums use subinterval midpoints.

Guided walkthrough

Three estimates for one integral

Estimate the area under $f(x)=x^2+1$ on $[0,2]$ using four rectangles.

Answer reveal

Worked solution

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Interactive checku3a-riemann-01

Use two left-endpoint rectangles to approximate $\int_0^2 (x+2)\,dx$ .

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Show hint

Each subinterval has width 1. Use heights $f(0)$ and $f(1)$ .

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**Visual study**
Three Riemann-sum choices. Compare left, right, and midpoint rectangles for the same increasing curve.

After the explanation

Use the section idea

Reading lens

Treat a definite integral as the limit of structured approximations, with the sample rule and sign visible in every rectangle.

Mental model

Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.

Decision

Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.

Common trap

Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.

Check yourself

Can you construct the sum from a table or formula and predict whether it is high or low before calculating?

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