Integrals

Evaluating Integrals with the FTC

Using the Fundamental Theorem of Calculus to evaluate definite integrals.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Antiderivatives

Evaluating a Definite Integral Without a Limit of Sums

Every definite integral evaluated so far has needed either a geometric shortcut or a full Riemann-sum limit. Before reading on: could an antiderivative — found without any limit process at all — evaluate a definite integral directly?

Definition — The Fundamental Theorem of Calculus

If F is any antiderivative of f (F' = f), then ∫ₐᵇ f(x) dx = F(b) − F(a). A related fact: for an accumulation function A(x) = ∫ₐˣf(t)dt, A'(x) = f(x) — the derivative of an accumulated-area function returns the original function being accumulated.

\int_a^b f(x)\,dx = F(b) - F(a), \quad \text{where } F' = f

Worked Example — Evaluating a Definite Integral with an Antiderivative

Evaluate ∫₁⁴ (4x³ − 6x + 5) dx, reusing the antiderivative F(x) = x⁴ − 3x² + 5x from the previous lesson. F(4) = 256 − 48 + 20 = 228. F(1) = 1 − 3 + 5 = 3. ∫₁⁴(4x³−6x+5)dx = F(4) − F(1) = 228 − 3 = 225.

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

Always verify F' actually equals the original integrand before using F in the Fundamental Theorem — a single missing or mis-signed term produces a wrong definite integral even if the rest of the evaluation is correct.

Common Mistakes

Computing F(a) − F(b) instead of F(b) − F(a).
The Fundamental Theorem's order is upper bound minus lower bound — reversing it negates the entire result.

Key Takeaways

The Fundamental Theorem evaluates a definite integral directly from any antiderivative: F(b) − F(a).
An accumulation function's derivative returns the original function — area and rate of change are two sides of the same relationship.

Summary

This closes the study of the integral and the Fundamental Theorem. The final unit applies definite integrals to genuine geometric and contextual questions.

Track your progress

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