The Timpson Lyceum · Calculus 1 (2024)

Fundamental Theorem of Calculus & Definite Integrals - Lecture 15 (Calculus 1)

Overview

This lecture develops the definite integral from its formal limit-of-Riemann-sums definition and then builds up to the central result of the course: the Fundamental Theorem of Calculus. We first nail down what $\int_a^b f(x)\,dx$ means as a number, see which functions are integrable, and collect the algebraic properties of definite integrals. We then prove the Mean Value Theorem for integrals (which guarantees a point where the function equals its average value) and use it to prove both parts of the Fundamental Theorem of Calculus — Part 1, that differentiating an accumulation function recovers the integrand, and Part 2, that a definite integral can be evaluated by any antiderivative. This matters because Part 2 turns area computation from a tedious limit of sums into a quick subtraction $F(b)-F(a)$.

Key Concepts

The definite integral (limit of Riemann sums)

• Partition $[a,b]$ into points $x_0=a, x_1, \dots, x_n=b$. On the $k$-th subinterval pick any sample point $c_k$ and form the Riemann sum $\sum_{k=1}^{n} f(c_k)\,\Delta x_k$, where $\Delta x_k = x_k - x_{k-1}$ is the width of that subinterval.
• The norm (or gap) of a partition, $\lVert P\rVert$, is the largest subinterval width.
• The definite integral is the limit of these sums as the norm shrinks to zero: $\int_a^b f(x)\,dx = \lim_{\lVert P\rVert \to 0} \sum_{k=1}^{n} f(c_k)\,\Delta x_k$ This limit is a single number $J$, independent of how the partition is refined or which sample points are chosen — exactly the $\varepsilon$–$\delta$ idea behind any limit. The variable ($x$, $t$, etc.) is a dummy variable: $\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$.
• Convenient special case: with $n$ equal subintervals of width $\frac{b-a}{n}$ and right-hand sample points $c_k = a + k\frac{b-a}{n}$, the integral becomes $\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n} f\!\left(a + k\tfrac{b-a}{n}\right)\frac{b-a}{n}$.

Integrability

• Theorem: if $f$ is continuous (or has at most finitely many jump discontinuities) on $[a,b]$, then $f$ is integrable on $[a,b]$.
• Not every function is integrable. The Dirichlet function $f(x)=1$ for rational $x$, $f(x)=0$ for irrational $x$, has upper sums equal to $b-a$ and lower sums equal to $0$ on any partition; the two never agree, so the limit fails to exist and $f$ is not integrable.

Properties of definite integrals (with $J=\int_a^b f(x)\,dx$)

• Reversing limits: $\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$.
• Zero width: $\int_a^a f(x)\,dx = 0$.
• Constant multiple: $\int_a^b k\,f(x)\,dx = k\int_a^b f(x)\,dx$.
• Sum/difference: $\int_a^b \big(f(x)\pm g(x)\big)\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$.
• Additivity over intervals: $\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$.
• Max–Min bound (Property 6): if $m$ and $M$ are the min and max of $f$ on $[a,b]$, then $m(b-a) \le \int_a^b f(x)\,dx \le M(b-a)$.
• Domination: if $f(x)\ge g(x)$ on $[a,b]$, then $\int_a^b f(x)\,dx \ge \int_a^b g(x)\,dx$.

Average value of a function

• The average (mean) value of $f$ on $[a,b]$ is the height of a rectangle on base $b-a$ with the same area as the region: $\text{avg}(f) = \frac{1}{b-a}\int_a^b f(x)\,dx$

Mean Value Theorem for definite integrals

• If $f$ is continuous on $[a,b]$, there exists a point $c \in [a,b]$ with $f(c) = \dfrac{1}{b-a}\int_a^b f(x)\,dx$. Proof idea: divide the Max–Min bound by $b-a$ to sandwich the average between $m$ and $M$, then apply the Intermediate Value Theorem.

Fundamental Theorem of Calculus, Part 1

• If $f$ is continuous and $F(x) = \displaystyle\int_a^x f(t)\,dt$ is its accumulation function, then $F$ is differentiable and $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$
• Intuition: the rate at which area accumulates is exactly the current height of the curve.
• With a variable upper limit $u(x)$, apply the chain rule: $\dfrac{d}{dx}\int_a^{u(x)} f(t)\,dt = f\big(u(x)\big)\,u'(x)$.

Fundamental Theorem of Calculus, Part 2 (Evaluation)

• If $F$ is any antiderivative of $f$ on $[a,b]$, then $\int_a^b f(x)\,dx = F(b) - F(a)$
• The arbitrary constant of antidifferentiation cancels in the subtraction, so any antiderivative works.

Applications introduced

• Displacement vs. distance: displacement $=\int v(t)\,dt$; total distance $=\int |v(t)|\,dt$ (split where $v$ changes sign so negative stretches are added, not subtracted).
• Area between curves: area $=\int_a^b |\text{top} - \text{bottom}|\,dx$; split at crossings since an integral returns a negative value where the region lies below the axis or where the lower curve is on top.

Worked Example

Evaluate $\displaystyle\int_1^4 \left(3\sqrt{x} - \frac{4}{x^2}\right)dx$ using FTC Part 2. Rewrite with exponents, antidifferentiate by the power rule $\big(\int x^n\,dx = \tfrac{x^{n+1}}{n+1}\big)$, then compute $F(4)-F(1)$.

Summary

• The definite integral $\int_a^b f(x)\,dx$ is defined as the limit of Riemann sums $\sum f(c_k)\,\Delta x_k$ as the partition norm goes to zero, and equals a single number when the limit is independent of the partition and sample points.
• Continuous functions (and those with finitely many jumps) are integrable; the Dirichlet function shows not everything is.
• Definite integrals obey clean rules: sign flips on reversed limits, zero over a point, linearity, additivity over adjacent intervals, the Max–Min bound, and domination.
• The average value $\frac{1}{b-a}\int_a^b f\,dx$ is actually attained by a continuous function at some interior point (Mean Value Theorem for integrals).
• FTC Part 1: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$ — differentiation undoes accumulation (with the chain rule for variable upper limits).
• FTC Part 2: $\int_a^b f(x)\,dx = F(b)-F(a)$ for any antiderivative $F$ — the practical tool for evaluating integrals, including displacement/distance and area-between-curves problems.