Calculus I · Adding Up the Infinite

Problems

Six tiers. Fluency and Practice are doing* — build Riemann sums and find antiderivatives until both are automatic (and like Chapter 4, there’s a lot of antiderivative drill, because it’s the same kind of mechanical skill). Concept & counterexample and ◆ Diamond are thinking — signed area, the $+C$, and the proofs. Teach it is teaching. Looking ahead opens Chapter 11. Do all of Fluency; antiderivative fluency is what makes the Fundamental Theorem usable.*

Fluency — do these fast

1. Antiderivatives, reverse power rule: (a) $\int x^4\,dx$; (b) $\int x^7\,dx$; (c) $\int x^{-2}\,dx$; (d) $\int \sqrt{x}\,dx$; (e) $\int 1\,dx$.
2. With constants and sums: (a) $\int (3x^2 - 2x + 5)\,dx$; (b) $\int (6x^5 + 4)\,dx$; (c) $\int \left(\tfrac1{x^2} + x\right)dx$.
3. Trig and exponential antiderivatives: (a) $\int \cos x\,dx$; (b) $\int \sin x\,dx$; (c) $\int e^x\,dx$; (d) $\int \sec^2 x\,dx$; (e) $\int \tfrac1x\,dx$.
4. Mixed library: (a) $\int (2\cos x - 3\sin x)\,dx$; (b) $\int (e^x + x^2)\,dx$; (c) $\int (4\sec^2 x - \tfrac5x)\,dx$.
5. Rewrite first, then integrate: (a) $\int \tfrac{x^2 - 1}{x}\,dx$ (split the fraction); (b) $\int x^2\sqrt{x}\,dx$ (one power); (c) $\int (x+1)^2\,dx$ (expand).
6. Simple substitution (the inside’s derivative is present): (a) $\int 2x(x^2+1)^5\,dx$; (b) $\int 3x^2(x^3+1)^4\,dx$; (c) $\int \cos x\,e^{\sin x}\,dx$; (d) $\int 4x^3(x^4+2)^6\,dx$; (e) $\int \sin^4 x\cos x\,dx$.
7. Substitution needing a constant factor, or landing on a log: (a) $\int x\,e^{x^2}\,dx$; (b) $\int x(x^2+1)^3\,dx$; (c) $\int \sin(3x)\,dx$; (d) $\int \cos(5x)\,dx$; (e) $\int \dfrac{x^2}{x^3+1}\,dx$ (the logarithm pattern).
8. Estimate $\int_0^4 f\,dx$ for $f(x) = x^2$ with a right Riemann sum using $n = 4$ rectangles (widths of $1$). Is your estimate above or below the true area, and why?
9. Evaluate by recognizing the area as a simple shape: (a) $\int_0^3 2\,dx$ (a rectangle); (b) $\int_0^2 x\,dx$ (a triangle); (c) $\int_{-1}^{1} x\,dx$ (watch the sign).

Practice — multi-step

1. Compute $\int_0^1 x^3\,dx$ from the definition, using $\sum_{i=1}^n i^3 = \left[\tfrac{n(n+1)}{2}\right]^2$ and a right Riemann sum. (Mirror the $\int_0^1 x^2$ computation in the chapter.)
2. Substitution: (a) $\int \dfrac{2x}{x^2+1}\,dx$; (b) $\int \dfrac{x}{\sqrt{x^2+9}}\,dx$; (c) $\int \sin x\cos x\,dx$ (let $u=\sin x$).
3. Substitution with rearrangement: (a) $\int \dfrac{\ln x}{x}\,dx$ (let $u = \ln x$); (b) $\int x\sqrt{x+1}\,dx$ (let $u = x+1$, then write $x = u-1$).
4. Find the function $F$ with $F'(x) = 3x^2 - 2$ and $F(1) = 5$. (Antiderivative, then pin down $C$ from the condition — your first “initial value problem.”)
5. A ball’s velocity is $v(t) = -10t + 20$ (m/s). Find its position $s(t)$ given $s(0) = 0$, by antidifferentiating. (You’re undoing the Chapter 3 derivative that produced $v$ from $s$.)
6. Set up — but do not yet evaluate — the area under $f(x) = e^{-x^2}$ from $0$ to $1$ as a limit of Riemann sums, and explain why you cannot finish it by finding an antiderivative.
7. Sketch $f(x) = x^2 - 1$ on $[0,2]$ and explain, using signed area, why $\int_0^2 (x^2-1)\,dx$ is smaller than the unsigned area between the curve and the axis.

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

1. 1. $\int_a^b f\,dx$ is a number; $\int f\,dx$ is a function. (b) If $F' = G'$, then $F = G$. (c) $\int_a^b f\,dx$ can be negative even when $f$ is a “real” function. (d) Every continuous function has an antiderivative that can be written with elementary functions.
2. Why the $+C$? Explain, citing the Mean Value Theorem, why two antiderivatives of the same function must differ by a constant — and why this means $\int f\,dx$ is a whole family.
3. Find the error. A student writes $\int \sin x\,dx = \cos x + C$. What’s wrong, and how would differentiating the answer have caught it?
4. Construct a function $f$ on $[-1,1]$ whose definite integral $\int_{-1}^1 f\,dx$ is $0$ even though $f$ is never zero. (Use signed area — what symmetry forces cancellation?)
5. Explain when substitution will and won’t help: why does it work for $\int 2x(x^2+1)^5\,dx$ but not, as written, for $\int (x^2+1)^5\,dx$? What’s missing in the second?
6. Is it possible? A definite integral $\int_a^b f\,dx$ that equals $0$ with $a < b$ and $f > 0$ somewhere on $[a,b]$. (What must $f$ do elsewhere?)

◆ Diamond — the definitions and the proofs

1. ◆ Compute $\int_0^1 x^2\,dx$ from the definition in full, deriving the limit of the right Riemann sum and using $\sum i^2 = \tfrac{n(n+1)(2n+1)}{6}$. (Reproduce the chapter’s computation, justifying each step.)
2. ◆ Prove the linearity of the definite integral, $\int_a^b (f+g) = \int_a^b f + \int_a^b g$, from the Riemann-sum definition. (A sum of sums is the sum of the sums; take the limit.)
3. ◆ Derive substitution from the chain rule. Starting from $\tfrac{d}{dx}F(g(x)) = F'(g(x))g'(x)$, integrate both sides to obtain the substitution rule $\int F'(g(x))g'(x)\,dx = F(g(x)) + C$, and explain why it is exactly the chain rule read backward.
4. ◆ Prove that a function with $F'(x) = 0$ for all $x$ on an interval is constant there (the fact behind the $+C$). (This is a one-line corollary of the Mean Value Theorem — state it.)
5. ◆◆ The midpoint sum is sharp. For $f(x) = x^2$ on $[0,1]$, compute the midpoint Riemann sum with $n$ rectangles and show it also converges to $\tfrac13$ — but, by comparing to the right sum, argue that it approaches the true value faster. (You’ll need $\sum (i - \tfrac12)^2$.)
6. ◆◆ Explain why $\int_a^b f\,dx$ being defined as a limit means it can fail to exist for badly behaved $f$ (for instance, a function that jumps infinitely often). What property of $f$ — met by every continuous function — guarantees the Riemann sums actually converge? (Name it; the proof is a real-analysis matter.)

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

1. In three or four sentences with the rectangles picture, teach a friend why the area under a curve is a limit of sums, and how thinner rectangles make the estimate exact.
2. Debug your classmate. They compute $\int 2x(x^2+1)^5\,dx = \tfrac{(x^2+1)^6}{6} \cdot 2x + C$ — they “kept the $2x$.” Explain where their substitution went wrong and give the correct answer.
3. Write a complete worked solution to $\int x\,e^{x^2}\,dx$, narrating the choice of $u$, the handling of the stray constant, and the back-substitution, so a peer could redo it for $\int x\sin(x^2)\,dx$.
4. Design a problem. Invent an integral that requires a substitution, choose it so the answer is clean, and write the full solution key (including the differentiate-to-check step).
5. In a few sentences, teach the difference between the definite integral (a number, an area) and the indefinite integral (a family of functions), and why they nonetheless share a symbol.

Looking ahead — Chapter 11, the summit, in your hands early

1. Define $A(x) = \int_0^x t^2\,dt$ — the area under $t^2$ from $0$ out to a movable right edge $x$. Using your result $\int_0^1 t^2\,dt = \tfrac13$, guess a formula for $A(x)$, then compute $A'(x)$. What do you notice about $A'(x)$ compared to the integrand $t^2$? (You are about to discover the Fundamental Theorem yourself.)
2. If finding the area under $f$ amounts to finding an antiderivative $F$ and looking at its endpoint values, predict the formula for $\int_a^b f\,dx$ in terms of $F(a)$ and $F(b)$. (Chapter 11 proves your guess and turns every area problem into a subtraction.)
3. The one antiderivative that escaped the power rule was $\int \tfrac1x\,dx = \ln|x|$. Predict what the function $L(x) = \int_1^x \tfrac1t\,dt$ has to do with the logarithm — in particular, what is $L(1)$, and is $L$ increasing or decreasing for $x > 1$? (Chapter 11 defines $\ln$ this way and repays every loan from Chapter 5.)

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