Chapter 14 — Exercises

40 problems on the Fundamental Theorem of Calculus, tiered ⭐ (routine) to ⭐⭐⭐⭐ (challenge). Work them with antiderivatives and the evaluation bar; reach for Python only to check, never to replace, a hand computation. Section references point back to index.md.

How to Use These Problems

Each problem is tagged with a star rating:

Answers to odd-numbered problems appear in appendices/answers-to-selected.md. Verify every hand computation against the methods in §14.4–§14.9 before checking.

Part A — Evaluating Definite Integrals via FTC Part 2 (§14.4, §14.6)

A1. ⭐ Evaluate $\displaystyle\int_0^3 x^2\,dx$.

A2. ⭐ Evaluate $\displaystyle\int_1^4 \sqrt{x}\,dx$.

A3. ⭐ Evaluate $\displaystyle\int_0^{\pi/2} \cos x\,dx$.

A4. ⭐ Evaluate $\displaystyle\int_1^2 \frac{1}{x}\,dx$.

A5. ⭐ Evaluate $\displaystyle\int_0^1 (4x^3 - 2x + 1)\,dx$.

A6. ⭐ Evaluate $\displaystyle\int_{-1}^{1} x^5\,dx$ and explain the answer using symmetry (§14.6, Example 4).

A7. ⭐⭐ Evaluate $\displaystyle\int_0^1 e^{2x}\,dx$.

A8. ⭐⭐ Evaluate $\displaystyle\int_0^{1} \frac{1}{1+x^2}\,dx$ and identify the constant it produces.

A9. ⭐⭐ Evaluate $\displaystyle\int_1^{e^2} \frac{1}{x}\,dx$.

A10. ⭐⭐ Evaluate $\displaystyle\int_0^{\pi/4} \sec^2 x\,dx$.

A11. ⭐⭐ Evaluate $\displaystyle\int_0^2 (3t^2 - 2t)\,dt$, then state its net-change interpretation if the integrand is a rate (§14.7).

A12. ⭐⭐⭐ Evaluate $\displaystyle\int_1^4 \frac{x - 1}{\sqrt{x}}\,dx$ by first splitting the integrand into a sum of powers.

Part B — Accumulation Functions and FTC Part 1 (§14.2, §14.3)

B1. ⭐ Let $F(x) = \displaystyle\int_2^x t^3\,dt$. Find $F'(x)$ without computing $F$.

B2. ⭐ Let $F(x) = \displaystyle\int_0^x \cos t\,dt$. Find $F'(x)$ and identify $F$ in closed form.

B3. ⭐⭐ Let $F(x) = \displaystyle\int_0^x f(t)\,dt$ with $f$ positive on $(0,2)$, zero at $x=2$, and negative on $(2,5)$. State where $F$ is increasing, where it is decreasing, and where it attains its maximum (§14.2, Check Your Understanding).

B4. ⭐⭐ Let $F(x) = \displaystyle\int_1^x \frac{1}{t}\,dt$ for $x > 0$. Compute $F'(x)$, identify $F$ in closed form, and explain why $F(1)=0$.

B5. ⭐⭐ Let $F(x) = \displaystyle\int_0^x (t-1)(t-3)\,dt$. Find the critical points of $F$ and classify each as a local max or min of $F$ using the sign of the integrand (FTC Part 1 plus the first-derivative test).

B6. ⭐⭐⭐ An accumulation $F(x) = \int_0^x f(t)\,dt$ has $F(0)=0$, where $f$ is the piecewise-linear graph rising from $f(0)=0$ to $f(2)=4$, then constant at $4$ on $[2,5]$. Compute $F(2)$ and $F(5)$ using areas, and confirm $F'(3)=4$ via FTC Part 1.

Part C — Differentiating Integrals with Variable Limits (§14.8)

C1. ⭐⭐ Compute $\dfrac{d}{dx}\displaystyle\int_0^{x^2} \sin t\,dt$.

C2. ⭐⭐ Compute $\dfrac{d}{dx}\displaystyle\int_1^{x^3} e^{-t^2}\,dt$. (Watch the chain-rule factor — §14.8 Common Pitfall.)

C3. ⭐⭐ Compute $\dfrac{d}{dx}\displaystyle\int_x^{5} \cos(t^2)\,dt$. (The variable is the lower limit.)

C4. ⭐⭐⭐ Compute $\dfrac{d}{dx}\displaystyle\int_{x}^{x^2} \frac{1}{1+t^2}\,dt$ using the two-limit Leibniz form.

C5. ⭐⭐⭐ Compute $\dfrac{d}{dx}\displaystyle\int_{\sin x}^{\cos x} e^{t}\,dt$ and simplify.

C6. ⭐⭐⭐ Let $G(x) = \displaystyle\int_0^{x^2} \sqrt{1 + t^3}\,dt$. Find $G'(2)$ as an exact number.

Part D — Net Change, Displacement, and Distance (§14.7)

D1. ⭐⭐ A particle has velocity $v(t) = 2t - 6$ m/s on $[0,5]$. Find its net displacement.

D2. ⭐⭐⭐ For the same particle in D1, find the total distance traveled on $[0,5]$. (Split at the sign change — §14.7, §14.14 Error 3.)

D3. ⭐⭐ Water flows into a tank at rate $r(t) = 4 + 6t$ liters/min on $[0,3]$. How many liters enter the tank?

D4. ⭐⭐ A population grows at rate $P'(t) = 200e^{0.1t}$ individuals/year. Find the net change in population over $[0,10]$. Leave the answer in exact form.

D5. ⭐⭐⭐ A car accelerates from rest with $a(t) = 12 - 3t$ m/s² on $[0,4]$. Find (a) its velocity at $t=4$ given $v(0)=0$, and (b) the distance it travels on $[0,4]$. (Two layers of net change.)

Part E — Average Value (§14.9)

E1. ⭐⭐ Find the average value of $f(x) = x^2$ on $[0,3]$ and the point $c$ where it is attained.

E2. ⭐⭐ Find the average value of $f(x) = \sin x$ on $[0, \pi]$.

E3. ⭐⭐⭐ The temperature of a room over a 12-hour shift is $T(t) = 68 + 4\sin\!\left(\frac{\pi t}{12}\right)$ °F. Find the average temperature over $[0,12]$.

Part F — Applications Across Fields (§14.7, §14.10; at least two fields, all ⭐⭐⭐+)

F1. ⭐⭐⭐ (Economics) A firm's marginal cost is $C'(q) = 0.03q^2 - 1.2q + 25$ dollars per unit. Find the added cost of increasing production from $q = 10$ to $q = 20$ units, and explain why fixed costs do not appear (§14.10).

F2. ⭐⭐⭐ (Physics — work) A spring obeys $F(x) = 80x$ newtons. Find the work $W = \int_0^{0.25} F(x)\,dx$ done stretching it from rest to $0.25$ m (§14.10).

F3. ⭐⭐⭐ (Medicine — pharmacokinetics) A drug is cleared from the blood at rate $r(t) = 30e^{-0.2t}$ mg/hr. Find the total mass cleared over the first 5 hours, $\int_0^5 r(t)\,dt$, in exact and decimal form (§14.10).

F4. ⭐⭐⭐ (Data science — probability) For the exponential density $p(x) = 2e^{-2x}$ on $[0,\infty)$, the CDF is the accumulation $F(x) = \int_0^x p(t)\,dt$. Compute $F(x)$ in closed form and find $P(0 \le X \le 1) = F(1)$ (§14.10).

Part G — Theory, Edge Cases, and the Limits of FTC (§14.4, §14.12) ⭐⭐⭐⭐

G1. ⭐⭐⭐⭐ A student writes $\displaystyle\int_{-1}^{1}\frac{1}{x^2}\,dx = \left[-\frac{1}{x}\right]_{-1}^{1} = -1 - 1 = -2$. Explain in one paragraph exactly why this is wrong, citing the continuity hypothesis of FTC Part 2 (§14.4 Common Pitfall), and name the chapter where the correct treatment lives.

G2. ⭐⭐⭐⭐ Prove that two antiderivatives of the same continuous function on an interval differ by a constant, and explain how that fact makes FTC Part 2 independent of which antiderivative you choose (§14.4, "Why it follows from Part 1").

G3. ⭐⭐⭐⭐ Let $F(x) = \displaystyle\int_0^x e^{-t^2}\,dt$. (a) Explain why $F$ has no elementary closed form (§14.12, Liouville). (b) Despite this, write $F'(x)$ exactly. (c) Argue that $F$ is strictly increasing.

G4. ⭐⭐⭐⭐ Define $H(x) = \displaystyle\int_0^{x} \lfloor t \rfloor\,dt$ where $\lfloor t \rfloor$ is the floor function, a step function with jumps at the integers. Compute $H(2.5)$ by summing rectangular areas, and explain why $H'(x)$ fails to exist at $x = 1$ and $x = 2$ even though $H$ is continuous there (§14.3 Warning).

A Note on Checking with Python

After you finish a problem by hand, you may confirm a numerical answer with scipy.integrate.quad, exactly as in §14.11. For F3, find the antiderivative by hand as $-150e^{-0.2t}\big|_0^5 = 150\!\left(1 - e^{-1}\right) \approx 94.82$ mg, then:

from math import e
from scipy.integrate import quad
val, _ = quad(lambda t: 30 * e**(-0.2*t), 0, 5)
print(val)   # ≈ 94.82  — matches the hand result 150(1 - e^{-1})

The machine confirms; it does not replace. The understanding is built by the antiderivative you found by hand.