Calculus I · The Fundamental Theorem of Calculus

Where this is heading

You have now built single-variable calculus from the ground up — limits, derivatives, the rules, the transcendental functions, the great theorems, the integral, and the Fundamental Theorem that binds the two halves into one — with every result derived and every loan repaid. Two doors stand open.

Chapter 12 stays in this book and finishes a thread from Chapter 1. We have added up infinitely many infinitesimal pieces (integrals); what about adding up infinitely many ordinary numbers — an infinite series? When does $1 + \tfrac12 + \tfrac14 + \cdots$ settle to a value and $1 + \tfrac12 + \tfrac13 + \cdots$ run off to infinity? The tool that decides it is the integral test, which weighs a sum against an area using the very picture this chapter is made of — and it will let us finally say, rigorously, where the Taylor series of Chapter 7 are allowed to live.

Book 3 opens the door to many dimensions: the derivative becomes the gradient, the single integral becomes a double and triple integral (volume measured exactly the way we measured area), optimization becomes the real landscape that machine learning descends, and the waves of Fourier finally get the orthogonal-basis home they were waiting for. The slope of a single point started all of this; it doesn’t stop here.

Problems

Six tiers. Fluency and Practice are doing* — evaluate definite integrals with the FTC and integrate by parts until both are automatic (plenty of drill, as in Chapter 4). Concept & counterexample and ◆ Diamond are thinking — the theorem itself and the proofs. Teach it is teaching. Looking ahead opens Chapter 12. The applications problems below ask you to set up the integral from the slicing principle — not to recall a formula. Do all of Fluency; FTC fluency is the working core of all of integral calculus.*

Fluency — evaluate with the FTC, fast

1. $\int_0^2 x^3\,dx$. 2. $\int_1^4 \sqrt{x}\,dx$. 3. $\int_0^{\pi} \sin x\,dx$. 4. $\int_0^{\pi/2}\cos x\,dx$. 5. $\int_1^e \tfrac1x\,dx$.
2. $\int_0^1 e^x\,dx$. 7. $\int_{-1}^{1}(x^2 - 1)\,dx$ (and say what the sign means). 8. $\int_1^2 (3x^2 - 2x)\,dx$. 9. $\int_0^{\pi/4}\sec^2 x\,dx$. 10. $\int_1^3 \left(x + \tfrac1x\right)dx$.
3. Use FTC Part 1 to differentiate without integrating: (a) $\tfrac{d}{dx}\int_0^x \sin(t^2)\,dt$; (b) $\tfrac{d}{dx}\int_2^x e^{-t^2}\,dt$; (c) $\tfrac{d}{dx}\int_1^x \tfrac{1}{1+t^4}\,dt$.
4. Integration by parts: (a) $\int x e^x\,dx$; (b) $\int x\cos x\,dx$; (c) $\int x\sin x\,dx$; (d) $\int x e^{-x}\,dx$; (e) $\int (x+1)\sin x\,dx$.
5. Integration by parts, harder: (a) $\int \ln x\,dx$; (b) $\int x e^{2x}\,dx$; (c) $\int x^2 e^x\,dx$ (parts twice); (d) $\int x^2\cos x\,dx$ (parts twice); (e) $\int x\ln x\,dx$ (take $u = \ln x$).

Practice — multi-step, and set up the slice

1. Evaluate $\int_0^1 x e^{x^2}\,dx$ two ways — by substitution, and by recognizing an antiderivative — and confirm they agree.
2. Find the area between $y = x$ and $y = x^2$ from $x=0$ to $x=1$. (Set up top $-$ bottom, then integrate.)
3. A region under $y = \tfrac1x$ from $x=1$ to $x=4$ is spun around the $x$-axis. Set up the volume by slicing into disks, then evaluate.
4. Set up (do not evaluate) the length of the parabola $y = x^2$ from $x=0$ to $x=1$ using the slicing principle, and explain why you should expect not to get a clean closed form.
5. Find the function $F$ with $F'(x) = \cos x$ and $F(0) = 3$. Then use it to compute $\int_0^{\pi} \cos x\,dx$ two ways (antiderivative-and-subtract, and directly), confirming they match.
6. Using $\ln x = \int_1^x \tfrac1t\,dt$, show directly from the integral that $\ln(2\cdot 3) = \ln 2 + \ln 3$, by performing the substitution in the chapter on $\int_2^{6}\tfrac1t\,dt$.

Concept & counterexample — think; true/false with justification, or build the example

1. 1. $\tfrac{d}{dx}\int_a^x f(t)\,dt = f(x)$. (b) $\int_a^b f'(x)\,dx = f(b) - f(a)$. (c) If $F$ and $G$ are both antiderivatives of $f$, then $\int_a^b f = F(b)-F(a) = G(b)-G(a)$. (d) Every continuous function has an antiderivative.
2. Explain, in your own words and citing FTC Part 1, why $\int_a^x f(t)\,dt$ is guaranteed to be an antiderivative of $f$ even when no elementary antiderivative of $f$ exists (as for $e^{-t^2}$).
3. Find the error. A student computes $\int_{-1}^{1}\tfrac{1}{x^2}\,dx = \left[-\tfrac1x\right]_{-1}^1 = -1 - 1 = -2$, a negative number for a positive function. What went wrong? (Where does $\tfrac1{x^2}$ fail the FTC’s hypotheses on $[-1,1]$?)
4. The two parts of the FTC say differentiation and integration undo each other. State precisely what “undo” means in each part — which operation is done first, and what you get back.
5. Is it possible? Two different functions $f \ne g$ with $\int_a^x f(t)\,dt = \int_a^x g(t)\,dt$ for all $x$. (Differentiate both sides — what must be true?)

◆ Diamond — the theorem and the capstone

1. ◆ Prove FTC Part 1 rigorously using the strip argument: bound $A(x+h) - A(x)$ between $m_h\cdot h$ and $M_h\cdot h$, divide by $h$, and squeeze as $h\to 0$ (citing the Extreme Value and Squeeze Theorems of Chapter 2).
2. ◆ Prove FTC Part 2 from Part 1, using the fact (Chapter 6) that two antiderivatives differ by a constant. Identify the exact step where $A(a) = 0$ is used.
3. ◆ Derive integration by parts from the product rule, and use it to evaluate $\int \ln x\,dx$, explaining the unusual choice $dv = dx$.
4. ◆◆ From the definition $\ln x = \int_1^x \tfrac1t\,dt$, prove the power law $\ln(x^r) = r\ln x$ for the case $r$ a positive integer. (Hint: use the product law repeatedly, or substitute.)
5. ◆◆ Prove that $e$, defined as the number with $\ln e = 1$, exists and is unique, citing the Intermediate Value Theorem (existence) and the fact that $\ln$ is strictly increasing (uniqueness). Explain how this discharges the loan first taken in Chapter 1.
6. ◆◆ Define $E(x)$ as the inverse function of $\ln$. Using the inverse-function rule (Chapter 4) and $\ln' x = \tfrac1x$, prove $E'(x) = E(x)$ — i.e., the inverse of the log-integral is its own derivative, which is why it deserves the name $e^x$.

Teach it — explain it, write it up, or fix it

1. In three or four sentences with the moving-edge picture, teach a friend why differentiating an area gives back the curve (FTC Part 1).
2. Debug your classmate. They evaluate $\int_0^{\pi}\sin x\,dx$ as $[\cos x]_0^\pi = \cos\pi - \cos 0 = -2$. Find both errors (the antiderivative and the sign) and give the correct value.
3. Write a complete worked solution to $\int x^2 e^x\,dx$ (integration by parts, applied twice), narrating why each step reduces the power, so a peer could do $\int x^3 e^x\,dx$.
4. In a few sentences, teach the slicing principle — how area-between-curves, volume, and arc length are all “what does one thin piece contribute, summed up” — and why that means you never need to memorize their formulas.
5. Teach the capstone. In a short paragraph, explain to someone who has finished Chapter 5 why defining $\ln x = \int_1^x \tfrac1t\,dt$ is more honest than just asserting $\ln' x = \tfrac1x$ — what does the definition give you that the assertion didn’t?

Looking ahead — Chapter 12 in your hands early

1. The integral $\int_1^\infty \tfrac1x\,dx$ is the area under $\tfrac1x$ stretching out forever. Using $\int_1^b \tfrac1x\,dx = \ln b$, decide whether this area is finite or infinite as $b\to\infty$. (You’ve just decided whether the series $1 + \tfrac12 + \tfrac13 + \cdots$ converges — Chapter 12 makes the link exact.)
2. Compare $\int_1^\infty \tfrac{1}{x^2}\,dx$ (compute it, as $b\to\infty$) with the previous problem. One area is finite, the other infinite. Predict which of $\sum \tfrac1{n^2}$ and $\sum \tfrac1n$ converges, and articulate the rule you’re sensing: a sum and the matching area live or die together. (That rule is Chapter 12’s integral test, and it is made of this chapter’s pictures.)

All 12 chapters of Calculus I · Foundations · All courses