Fundamental Theorem of Calculus Part 2¶

Theorem¶

Suppose $a < b$ are real numbers, and let $f\colon [a, b] \to\mathbb{R}$ be a Riemann integrable function. If $F\colon [a, b] \to\mathbb{R}$ is an antiderivative of $f\text{,}$ then

\int_a^b f(x)\,\mathrm{d}x =F(x)\big|_{x=a}^{b}=F(b)-F(a).

Interpretation¶

This part of the fundamental theorem turns the evaluation of a definite integral into a subtraction. Once an antiderivative $F$ of $f$ is known, the limit of Riemann sums equals $F(b)-F(a)$ $F (a)$ , and no sums have to be computed.

Two panels: the area under f from a to b, and the graph of an antiderivative F whose rise from F(a) to F(b) equals that area.

The left panel shows the area under $f$ between $a$ and $b$ ; the right panel shows an antiderivative $F$ , which climbs from $F(a)$ to $F(b)$ . The theorem says that these are the same number: the area accumulated under $f$ equals the total rise of $F$ . Where $f$ is large, $F$ climbs steeply, and where $f$ is small, $F$ is nearly flat.

The reason is a telescoping sum. For a partition $P$ , the mean value theorem gives points $c_i$ with $F(x_i)-F(x_{i-1})=f(c_i)(x_i-x_{i-1})$ $x_{i - 1})$ . With these tags the Riemann sum is exactly $F(b)-F(a)$ $F (a)$ , for every $P$ , so the integral is $F(b)-F(a)$ as well.

Both hypotheses are needed. The integral must exist: $F(x)=x^2\sin(1/x^2)$ with $F(0)=0$ is differentiable on $[0,1]$ , but $F'$ is unbounded near $0$ and therefore not Riemann integrable. And $F$ must be an antiderivative on all of $[a,b]$ . The function $-\arctan(1/x)$ has derivative $\frac{1}{1+x^2}$ for $x\ne0$ $0$ , yet its difference between $x=1$ and $x=-1$ is $-\frac{\pi}{2}$ , while $\int_{-1}^{1}\frac{\mathrm{d}x}{1+x^2}=\frac{\pi}{2}$ $\frac{π}{2}$ ; it jumps at $0$ , so the theorem does not apply to it.

Later, the rules of integration by parts and substitution combine this theorem with the product rule and the chain rule.

Key Results¶

Definite integrals can be evaluated using antiderivatives.
The result depends only on boundary values.
Integration becomes an algebraic operation once antiderivatives are known.

Examples¶

$\displaystyle \int_0^1 x^2\,\mathrm{d}x=\frac{1}{3}\text{.}$
$\displaystyle \int_a^b 1\,\mathrm{d}x=b-a\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 11

Aliases: FTC 2, second fundamental theorem of calculus, Newton-Leibniz formula, Newton-Leibniz theorem Symbols: integral, pi, real numbers

Last updated: 20 October 2026 03:02