Calculus I · Unit 3A · lesson
Left, Right, and Midpoint Riemann Sums
Learn Left, Right, and Midpoint Riemann Sums through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Riemann sums and the definite integralWhat this section is building
Learn Left, Right, and Midpoint Riemann Sums through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.
Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.
Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.
Learning objectives
Construct common Riemann sums, calculate them from formulas or tables, and reason about overestimates and underestimates.
Left, Right, and Midpoint Riemann Sums
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 . The rectangle contribution is
Adding gives
Left sums use left endpoints, right sums use right endpoints, and midpoint sums use subinterval midpoints.
Three estimates for one integral
Estimate the area under on using four rectangles.
Worked solution
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u3a-riemann-01Use two left-endpoint rectangles to approximate .
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Each subinterval has width 1. Use heights and .
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Read this graph as text
Three Riemann-sum choices. An increasing curved graph with rectangles whose heights use left, right, or midpoint samples. Compare left, right, and midpoint rectangles for the same increasing curve. Do not imply midpoint is always exact or always an overestimate.
Every relationship in three riemann-sum choices uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Compare left, right, and midpoint rectangles for the same increasing curve.
Three Riemann-sum choices. Compare left, right, and midpoint rectangles for the same increasing curve.
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