Fundamental Theorem of Calculus Part 1¶

Theorem¶

Suppose $a < b$ are real numbers, $f\colon [a, b] \to\mathbb{R}$ is a Riemann integrable function, and $F\colon [a, b] \to\mathbb{R}$ is the function defined as:

F(x)=\int_a^x f(t)\,\mathrm{d}t.

Then, $F$ is continuous on $[a,b]\text{.}$ Moreover, if $x_0\in(a,b)$ and $f$ is continuous at $x=x_0\text{,}$ then $F$ is differentiable at $x=x_0\text{,}$ with $F'(x_0)=f(x_0)\text{.}$

Interpretation¶

Integrating $f$ from $a$ up to a moving point $x$ produces a new function $F$ , and this part of the fundamental theorem describes how $F$ changes: it is always continuous, and where $f$ is continuous its rate of change is the current value $f(x)$ . In this sense differentiation undoes integration.

The area under a curve from a to x shaded as F(x), and a thin extra strip from x to x + h of area about f(x) times h, showing that F'(x) = f(x).

The figure shows why. Moving the right edge from $x$ to $x+h$ adds the thin red strip, whose area $F(x+h)-F(x)$ is about $f(x)\,h$ for small $h$ . Dividing by $h$ and letting $h\to0$ gives $F'(x)=f(x)$ $f (x)$ . The approximation becomes exact in the limit because $f$ is continuous at $x$ , so its values on the strip are all close to $f(x)$ .

The two conclusions need different hypotheses. Continuity of $F$ needs only integrability: a Riemann integrable $f$ is bounded, say $|f|\le M$ $M$ , and then $|F(x)-F(y)|\le M|x-y|$ $y ∣$ . The derivative statement needs continuity of $f$ at $x_0$ . Where $f$ jumps, $F$ has a corner and is not differentiable. Even a change at a single point matters: if $f(1)=1$ and $f(x)=0$ otherwise on $[0,2]$ , then $F$ is identically $0$ , so $F'(1)=0\ne f(1)$ $f (1)$ .

The theorem shows that every continuous function on an interval has an antiderivative, namely $F$ , and part 2 then turns integrals into differences $F(b)-F(a)$ $F (a)$ .

Key Results¶

Integration defines a continuous function.
Differentiation recovers the original integrand at points of continuity.
The theorem establishes a local connection between integration and differentiation.

Examples¶

If $f(x)=x^2\text{,}$ then $\displaystyle F(x)=\int_0^x t^2\,\mathrm{d}t$ and $F'(x)=x^2\text{.}$
For variable limits, suppose $g$ and $h$ are differentiable at $x$ , their values stay in $[a,b]$ near $x$ , and $f$ is continuous at both $g(x)$ and $h(x)$ . Then:

$\frac{\mathrm{d}}{\mathrm{d}x}\int_{g(x)}^{h(x)}f(t)\,\mathrm{d}t=h'(x)f(h(x)) -g'(x)f(g(x)).$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 11

Aliases: FTC 1, first fundamental theorem of calculus Symbols: element of, integral, real numbers

Last updated: 22 October 2026 19:04