Mathfolis

Riemann Sums

Unit 6 · Integration & Accumulation

Riemann Sums

A Riemann sum approximates the area under a curve using rectangles. The definite integral is the limit of these sums as the number of rectangles approaches infinity. Whether a Riemann sum overestimates or underestimates depends on the monotonicity and concavity of f.

Riemann Sum (nn equal subintervals, Δx=(b−a)/n\Delta x = (b-a)/n)

Left (LRAM)
∑i=0n−1f(xi) Δx\sum_{i=0}^{n-1} f(x_i)\,\Delta x
Right (RRAM)
∑i=1nf(xi) Δx\sum_{i=1}^{n} f(x_i)\,\Delta x
Trapezoidal
Δx2[f(x0)+2f(x1)+⋯+2f(xn−1)+f(xn)]\frac{\Delta x}{2}\bigl[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)\bigr]

Over/Underestimate Rules

f increasing: LRAM underestimates, RRAM overestimates
f decreasing: LRAM overestimates, RRAM underestimates
f concave up: Trapezoidal overestimates; f concave down: underestimates
AP Tip: The midpoint Riemann sum (MRAM) is usually the most accurate of the three rectangle methods.
Type 1

Left and Right Riemann Sums

Compute LRAM or RRAM using left or right endpoints of each subinterval, then determine whether the estimate is high or low.

Example 1
Approximate ∫₀³ (x² + 1) dx using LRAM with n = 3 subintervals. Is it an overestimate or underestimate?
Example 2
Use f(x) = 1/x on [1, 5] with n = 4. Compute LRAM and RRAM and classify each.

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Type 2

Trapezoidal Sum from a Table

Use tabular data with the trapezoidal rule. The subintervals need not be equal.

Example 3
The table gives f at x = 0, 2, 4, 6. Approximate ∫₀⁶ f(x) dx using the Trapezoidal Rule.

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Type 3

Comparing Approximation Methods

Given information about monotonicity and concavity, rank LRAM, RRAM, and Trapezoidal sum relative to the exact value.

Example 4
f is positive, increasing, and concave down on [1, 5]. Rank LRAM, Trapezoidal sum, and RRAM from smallest to largest.

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Riemann Sums | AP Calculus AB — Mathfolis