Chapter 14 — Key Takeaways

The one-paragraph version: differentiation and integration are inverse processes. FTC Part 1 says differentiating an accumulation function returns the integrand; FTC Part 2 says integrating a function is a subtraction of antiderivative values. Everything below is detail on those two sentences.

The Accumulation Function (§14.2)

  • Freeze the lower limit and let the upper limit move: $F(x) = \displaystyle\int_a^x f(t)\,dt$.
  • $F(x)$ is the running total of signed area swept from $a$ to $x$. Where $f > 0$, $F$ rises; where $f < 0$, $F$ falls; where $f = 0$, $F$ levels off.
  • The integration variable $t$ is a dummy; the value depends only on $x$, $a$, and $f$.
  • Mental model: if $f$ is a rate, $F$ is the accumulated amount (velocity → displacement, flow rate → volume, density → probability).

FTC Part 1 — Differentiation of an Integral (§14.3)

If $f$ is continuous near $a$ and $F(x) = \displaystyle\int_a^x f(t)\,dt$, then $F'(x) = f(x)$.

  • In words: the derivative of the accumulation function is the integrand. Differentiation undoes integration.
  • Existence payoff: every continuous function has an antiderivative — namely its own accumulation function. This is a powerful existence theorem hiding inside a formula.
  • Continuity is required. At a jump discontinuity of $f$, the accumulation $F$ is still continuous but has a corner, so $F'$ fails to exist exactly where $f$ jumps (§14.3 Warning).

FTC Part 2 — The Evaluation Theorem (§14.4)

If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$, then $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$.

  • The infinite limit of Riemann sums collapses to a single subtraction.
  • Any antiderivative works: two antiderivatives differ by a constant, and that constant cancels in $F(b) - F(a)$. This is why the "$+C$" never appears in a definite integral.
  • Continuity on the entire $[a,b]$ is required. A discontinuity inside the interval invalidates the evaluation bar — that is an improper integral (Chapter 17), not a routine FTC computation.

The Evaluation Bar (§14.5)

$$\int_a^b f(x)\,dx = F(x)\Big|_a^b = \Big[F(x)\Big]_a^b = F(b) - F(a).$$ Plug in the top, plug in the bottom, subtract bottom from top. Top limit first, always.

The Net Change Theorem (§14.7)

$\displaystyle\int_a^b F'(x)\,dx = F(b) - F(a)$: the integral of a rate of change is the net change.

  • Velocity integrates to net displacement; marginal cost to added cost; growth rate to net population change; probability density to interval probability.
  • Net displacement vs. total distance: $\int_a^b v\,dt$ is net change (legs can cancel); total distance is $\int_a^b |v|\,dt$, which requires splitting at the sign changes of $v$ (§14.14 Error 3).

Differentiating Integrals with Variable Limits (§14.8)

  • Variable upper limit + chain rule: $\dfrac{d}{dx}\displaystyle\int_a^{u(x)} f(t)\,dt = f(u(x))\,u'(x)$.
  • Both limits variable (Leibniz rule): $\dfrac{d}{dx}\displaystyle\int_{u(x)}^{v(x)} f(t)\,dt = f(v(x))\,v'(x) - f(u(x))\,u'(x)$.
  • Never drop the chain-rule factor — the derivative of the limit must multiply in.

Average Value (§14.9)

  • $\displaystyle \bar f = \frac{1}{b-a}\int_a^b f(x)\,dx$ — the height of the rectangle on $[a,b]$ with the same area as the region under $f$.
  • The Mean Value Theorem for Integrals guarantees a point $c \in [a,b]$ where $f(c) = \bar f$ is actually achieved.
  • Via FTC, $\bar f = \dfrac{F(b) - F(a)}{b - a}$ — the average slope of any antiderivative across $[a,b]$.

What FTC Does Not Promise (§14.12)

  • FTC Part 2 says if you have an antiderivative, the integral is a subtraction. It does not promise the antiderivative is elementary.
  • Liouville's theorem (1830s): $\int e^{-x^2}\,dx$, $\int \sin(x^2)\,dx$, $\int \frac{1}{\ln x}\,dx$ have no elementary antiderivative.
  • FTC still holds — we just evaluate numerically or by Taylor series (Chapter 23) instead of by formula.

Common Errors (§14.14)

  1. Ignoring a discontinuity inside the interval — check continuity on all of $[a,b]$ first (else Chapter 17).
  2. Dropping the chain-rule factor on variable limits (§14.8).
  3. Confusing net displacement with total distance — use $|v|$ for distance (§14.7).
  4. Reversing the subtraction — it is $F(\text{top}) - F(\text{bottom})$, not the reverse.
  5. Carrying "$+C$" into a definite integral — it cancels; never write it.

Connections — Why FTC Is "Fundamental" (§14.13)

  • Foundational, unifying, computational all at once: it binds the derivative (Chapter 6) and the definite integral (Chapter 13) into one structure and turns Riemann-sum limits into routine subtraction.
  • The seed of every higher integral theorem, all sharing the slogan the integral of a derivative over a region equals values on the boundary:
  • Fundamental Theorem for Line Integrals, $\int_C \nabla f\cdot d\mathbf{r} = f(B) - f(A)$ (Chapter 35);
  • Green's Theorem (Chapter 35);
  • Stokes' Theorem and the Divergence Theorem (Chapter 37);
  • the grand unification $\int_{\partial M}\omega = \int_M d\omega$ (Chapter 38).

Looking Ahead

Chapter 15 develops $u$-substitution and integration by parts — the techniques for finding the antiderivatives FTC consumes. Chapter 17 treats the improper integrals we deliberately mishandled in the §14.4 pitfall. Chapter 18 unleashes FTC on areas, volumes, arc length, and work. From here on, FTC is the floor every chapter stands on.