---
title: "The Fundamental Theorem of Calculus"
description: "Both halves: differentiating an accumulation function returns the integrand, and a definite integral can be evaluated from any antiderivative. This is why the two calculus operations undo each other."
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source: https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A.md
retrieved: 2026-09-12
---

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# The Fundamental Theorem of Calculus

Both halves: differentiating an accumulation function returns the integrand, and a definite integral can be evaluated from any antiderivative. This is why the two calculus operations undo each other.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A

## Ready when they can

- Differentiate an integral with a variable upper limit
- Evaluate a definite integral by antidifferentiation and justify the step
- Explain why an accumulation function is continuous even when the integrand is not

## Lesson: How accumulation and slope undo each other

Part 1 builds an accumulation function g of x as the integral from a to x of f of t dt, then says its derivative is f of x. Differentiating the buildup returns the integrand with x in place of t, so the integral of sin from 0 to x differentiates back to sin of x.

**Example.** Part 2 evaluates integrals from antiderivatives: find any F with derivative f, then the integral from a to b is F of b minus F of a. For 3x squared plus 1 from 0 to 2, use x cubed plus x: at 2 it is 10, at 0 it is 0, so the integral is 10. Added constants cancel in the subtraction, so any antiderivative works.

When the top limit is a function of x, pair the theorem with the chain rule. For the integral from 0 to x squared of cos, evaluate at the top to get cos of x squared, then multiply by the top derivative 2x, giving 2x cos of x squared. From a graph, f positive means g climbs, since g prime is f.

**Tip.** Accumulation smooths jumps: if f jumps at a point, g stays continuous there because the extra area over a tiny interval is tiny, though g may lose differentiability at the jump.

**Recap.** Differentiate accumulation to recover f, and evaluate integrals by subtracting one antiderivative at the limits.

## Practice

20 questions on this page, each with its working shown.

## Needs first

- [Integration as the Reverse of Differentiation](https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk)
- [Riemann Sums and the Definite Integral as a Limit](https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9)

## Opens up

- [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7)
- [Continuous Random Variables and Density Functions](https://lightmysky.com/learn/mathematics/continuous-random-variables-and-density-functions-mt_jlf4zDTeNm)
- [Stokes' Theorem and the Divergence Theorem](https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q)