Unit 6: Integration & Accumulation of Change

Riemann Sums & the Definite Integral

Approximating area with Riemann sums and defining the definite integral.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

From Approximating Area to Defining It Exactly

Definition — Riemann Sum and the Definite Integral

A Riemann sum approximates the area under f on [a, b] by summing the areas of n rectangles — left, right, or midpoint, depending on which point of each subinterval sets the height. The definite integral ∫ₐᵇf(x)dx is defined as the limit of such a sum as n → ∞: the exact net signed area, positive above the x-axis and negative below.

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\, \Delta x

Worked Example — Comparing Approximations to the Exact Value

Approximate ∫₀³ (x² + 1) dx with n = 3 (Δx = 1). With f(0)=1, f(1)=2, f(2)=5, f(3)=10: left sum L = 1(1+2+5) = 8; right sum R = 1(2+5+10) = 17; midpoint sum, using f(0.5)=1.25, f(1.5)=3.25, f(2.5)=7.25, gives M = 11.75. The exact value: ∫₀³(x²+1)dx = [x³/3+x]₀³ = 12. As expected for an increasing, concave-up function, L underestimates, R overestimates, and M (11.75) lands close to and slightly under the exact value.

Integral Visualizer

f(x) =

x^2

Integration bounds

a — lower bound0

b — upper bound2

Animate b (the play button above) to watch the shaded area — and the accumulation curve below — grow.

Approximation method

Method

Subdivisions (n)10

Show approximationShow exact area

Trace mode

f(x), with the region between a and b shaded

Show accumulation function A(x)

Accumulation function A(x) = ∫ₐˣ f(t) dt — linked to the graph above; panning or zooming either moves both

What the Fundamental Theorem of Calculus tells us

A(x) = ∫ from a to x of f(t) dt is the accumulation function. At x = b ≈ 2, A(b) ≈ 2.67 — the signed area under f from a ≈ 0 to b.
The Fundamental Theorem of Calculus says A'(x) = f(x): differentiating the accumulation function gives the original function back.
Confirmed numerically here: differentiating the accumulation curve gives A'(b) ≈ 4, matching f(b) ≈ 4.

Numerical approximation comparison

Every method's estimate of ∫ₐᵇ f(x) dx at n = 10 subdivisions, against a high-resolution reference — all should converge toward the same value.
Method	Estimate
Left Riemann sum	2.28
Right Riemann sum	3.08
Midpoint Riemann sum	2.66
Trapezoidal Rule	2.68
Simpson's Rule	2.66667
Exact (high-resolution reference)	2.66667

Analysis of f(x)

y-intercept: (0, 0)
x-intercepts: (0, 0)
Extrema: local min at (0, 0)
Inflection points: none found in view
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [0, 100]

Tip

Monotonicity predicts whether a left or right sum over- or underestimates; concavity predicts the same for the midpoint sum — both without computing the exact integral.

Common Mistakes

Treating a definite integral as always a positive, physical area.
A definite integral is signed area — where f(x) < 0, that portion subtracts from the total.

Key Takeaways

A Riemann sum approximates area with finitely many rectangles; the definite integral is its exact limit as n → ∞.
A definite integral measures net signed area, not necessarily a positive physical area.

Summary

The next lesson connects the definite integral directly to antiderivatives, and develops the main technique for finding them.

Track your progress

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