The Timpson Lyceum · Real Analysis (2020)

Lecture 22 Part 1 (Real Analysis 2020)

Overview

This lecture builds toward the Second Fundamental Theorem of Calculus, the counterpart to the first. Where the first theorem evaluates the integral of a derivative ( $\int_a^b f'(x)\,dx = f(b) - f(a)$ ), the second says that differentiating an integral recovers the integrand: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$ . To reach it we first prove two supporting results — the Mean Value Theorem for Integrals and the continuity of the integral (accumulation) function $F(x) = \int_a^x f(t)\,dt$ . These supply the geometric intuition (a rectangle whose area matches the area under the curve) and the analytic machinery (sandwiching, the Intermediate Value Theorem, uniform continuity) needed for the main proof.

Key Concepts

Tools we rely on

Extreme Value Theorem: a continuous function on a closed interval $[a,b]$ attains a minimum and a maximum — there exist points $x_m, x_M$ with $f(x_m) \le f(x) \le f(x_M)$ for all $x \in [a,b]$ .
Intermediate Value Theorem (IVT): a continuous function takes every value between any two of its attained values.
Uniform continuity (sequential view): $F$ is uniformly continuous if, for any points $u_n, v_n$ with $|u_n - v_n| \to 0$ , we have $|F(u_n) - F(v_n)| \to 0$ . On a closed bounded interval, uniform continuity implies (and is implied by) continuity.

Mean Value Theorem for Integrals

If $f$ is continuous on $[a,b]$ , there exists $x_0 \in [a,b]$ with $f(x_0) = \frac{1}{b-a}\int_a^b f(x)\,dx.$
Equivalently, $f(x_0)\,(b-a) = \int_a^b f(x)\,dx$ : there is a height $f(x_0)$ whose rectangle over $[a,b]$ has the same area as the region under the curve. This is the integral analogue of the average value.
Why it holds: by the Extreme Value Theorem $m \le f(x) \le M$ (with $m = f(x_m)$ , $M = f(x_M)$ ). Integrating, $m(b-a) \le \int_a^b f \le M(b-a)$ ; dividing by the positive number $b-a$ gives $m \le \frac{1}{b-a}\int_a^b f \le M$ . Since this value lies between two values $f$ attains, the IVT produces the point $x_0$ .

The accumulation function $F(x) = \int_a^x f(t)\,dt$

The variable $t$ is just a dummy of integration; $x$ is the upper limit and controls how far we integrate. $F$ depends only on $x$ . Geometrically $F(x)$ is the signed area from $a$ to $x$ .
$F$ is well defined: $f$ integrable on $[a,b]$ means $\int_a^x f$ exists for every $x \in [a,b]$ .
$F$ is continuous: since $f$ is bounded, pick $M > 0$ with $-M \le f(t) \le M$ . For points $u < v$ in $[a,b]$ , split $F(v) = \int_a^v f = \int_a^u f + \int_u^v f = F(u) + \int_u^v f$ , so $F(v) - F(u) = \int_u^v f$ . Bounding the integral by its width gives $|F(v) - F(u)| \le M\,(v - u).$ As $|v - u| \to 0$ the right side $\to 0$ , so $F$ is uniformly continuous, hence continuous.

Second Fundamental Theorem of Calculus

If $f$ is continuous on $[a,b]$ and $F(x) = \int_a^x f(t)\,dt$ , then $F$ is differentiable and $F'(x) = \frac{d}{dx}\int_a^x f(t)\,dt = f(x) \quad \text{for all } x \in [a,b].$
Differentiating with respect to the upper limit recovers the integrand evaluated at that limit.

Worked Example

Proof sketch of the Second Fundamental Theorem. Fix $x_0 \in [a,b]$ and form the difference quotient of $F$ . The numerator collapses to a single integral, the Mean Value Theorem for Integrals supplies a point $c(x)$ between $x_0$ and $x$ , and continuity finishes it as $x \to x_0$ :

\begin{aligned} F'(x_0) &= \lim_{x \to x_0} \frac{F(x) - F(x_0)}{x - x_0} \\ &= \lim_{x \to x_0} \frac{1}{x - x_0}\left(\int_a^x f(t)\,dt - \int_a^{x_0} f(t)\,dt\right) \\ &= \lim_{x \to x_0} \frac{1}{x - x_0}\int_{x_0}^{x} f(t)\,dt \\ &= \lim_{x \to x_0} f\big(c(x)\big), \qquad c(x) \in [x_0, x] \\ &= f(x_0). \end{aligned}

The third line uses additivity of the integral ( $\int_a^x - \int_a^{x_0} = \int_{x_0}^x$ ); the fourth uses the Mean Value Theorem for Integrals, which gives $c(x)$ with $\frac{1}{x-x_0}\int_{x_0}^x f = f(c(x))$ ; the last uses that $c(x)$ is trapped between $x_0$ and $x$ , so $c(x) \to x_0$ , and continuity of $f$ then forces $f(c(x)) \to f(x_0)$ . (Approaching from above is taken for concreteness; the one-sided argument from below is identical up to signs.)

Summary

The Second Fundamental Theorem is the mirror image of the first: differentiating an integral with a variable upper limit returns the integrand, $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$ .
The Mean Value Theorem for Integrals guarantees a height $f(x_0)$ whose rectangle over $[a,b]$ matches the area under the curve; it follows from the Extreme Value Theorem (sandwiching) plus the IVT.
The accumulation function $F(x) = \int_a^x f(t)\,dt$ measures signed area up to $x$ ; here $t$ is only a dummy variable and $F$ depends solely on $x$ .
$F$ is continuous (in fact uniformly continuous) because $|F(v) - F(u)| \le M|v - u|$ whenever $f$ is bounded by $M$ .
The proof of the main theorem chains three ideas: rewrite the difference quotient as a single integral $\int_{x_0}^x f$ , apply the Mean Value Theorem for Integrals to get $f(c(x))$ , then let $x \to x_0$ and use continuity.