Chapter 13 — Exercises

38 problems on the area problem, Riemann sums, the definite integral as a limit, signed area, properties, average value, and numerical integration. Difficulty runs ⭐ (recall and one-step) to ⭐⭐⭐⭐ (multi-step or proof).

These problems build the definite integral the honest way — by Riemann sums, geometry, and limits. Do not use the Fundamental Theorem of Calculus or antiderivatives; that shortcut is the subject of Chapter 14. Every value here is obtainable from sums (§13.3–13.5), signed area and geometry (§13.6–13.7), the properties of §13.8, the average-value tools of §13.11, or a numerical rule from §13.13.

Star ratings mark difficulty:

• ⭐ routine — direct application of one definition or formula
• ⭐⭐ standard — combines two ideas or requires a short setup
• ⭐⭐⭐ challenging — multi-step, careful bookkeeping, or a limit
• ⭐⭐⭐⭐ deep — synthesis, proof, or open-ended modeling judgment

Useful summation formulas (quote freely): $$\sum_{i=1}^n 1 = n,\qquad \sum_{i=1}^n i = \frac{n(n+1)}{2},\qquad \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6},\qquad \sum_{i=1}^n i^3 = \left[\frac{n(n+1)}{2}\right]^2.$$

Part A — Riemann Sums: Left, Right, and Midpoint (§13.2–13.3)

A1. ⭐ For $f(x) = 2x + 1$ on $[0, 4]$ with $n = 4$ subintervals ($\Delta x = 1$), compute the left Riemann sum $L_4$. List your four sample points before summing.

A2. ⭐ For the same $f(x) = 2x+1$ on $[0,4]$ with $n=4$, compute the right Riemann sum $R_4$.

A3. ⭐ For the same $f$ and partition, compute the midpoint sum $M_4$ (sample points $0.5, 1.5, 2.5, 3.5$). Compare $M_4$ to $L_4$ and $R_4$ from A1–A2.

A4. ⭐ Let $f(x) = x^2$ on $[1, 3]$ with $n = 4$. Find $\Delta x$ and list the right endpoints $x_1, x_2, x_3, x_4$. (Do not yet sum.)

A5. ⭐⭐ Continue A4: compute $R_4$ and $L_4$ for $\int_1^3 x^2\,dx$. Which one overestimates and which underestimates? Justify using the monotonicity of $f$ on $[1,3]$ (§13.2).

A6. ⭐⭐⭐ A function $f$ is measured at evenly spaced points on $[0, 6]$ (so $\Delta x = 1.5$, $n=4$):

Compute $L_4$, $R_4$, and the trapezoidal estimate $T_4$ of $\int_0^6 f\,dx$ directly from the table. Verify that $T_4 = \tfrac12(L_4 + R_4)$ (§13.13).

Part B — Sigma Notation (§13.3)

B1. ⭐ Evaluate $\displaystyle\sum_{i=1}^{5} (3i - 2)$ by writing out the terms.

B2. ⭐ Evaluate $\displaystyle\sum_{i=1}^{6} i^2$ using the sum-of-squares formula.

B3. ⭐ Rewrite the sum $1 + 4 + 9 + 16 + \cdots + 100$ in sigma notation, then evaluate it.

B4. ⭐⭐ Use the summation formulas to find a closed form for $\displaystyle\sum_{i=1}^{n} (4i + 3)$ as a function of $n$.

B5. ⭐⭐ Write the right-endpoint Riemann sum for $\int_2^5 (x^2 + 1)\,dx$ with $n$ subintervals in sigma notation. Identify $\Delta x$ and $x_i$ explicitly; do not take the limit (§13.3).

B6. ⭐⭐⭐ Write the midpoint Riemann sum for $\int_0^2 e^{-x}\,dx$ with $n$ subintervals in sigma notation, giving $\overline{x}_i = a + (i - \tfrac12)\Delta x$ explicitly. (Leave it as a sum — this integrand is the kind §13.13 and §13.14 say we approximate numerically.)

Part C — The Integral as a Limit of Sums (§13.4–13.5)

C1. ⭐ State, in one sentence and a formula, the definition of $\int_a^b f(x)\,dx$ as a limit of Riemann sums (§13.4). Identify which factor is the integrand and which is the limit of $\Delta x$.

C2. ⭐⭐ Evaluate $\displaystyle\int_0^1 x\,dx$ from the definition using right endpoints and $\sum_{i=1}^n i = \frac{n(n+1)}{2}$. Confirm your answer geometrically as a triangle (§13.5).

C3. ⭐⭐ Evaluate $\displaystyle\int_0^3 2x\,dx$ from the definition using right endpoints. Confirm geometrically.

C4. ⭐⭐⭐ Evaluate $\displaystyle\int_0^2 x^2\,dx$ from the definition using right endpoints. Show every step: $\Delta x = 2/n$, $x_i = 2i/n$, the sum-of-squares substitution, and the limit (model your work on §13.5).

C5. ⭐⭐⭐⭐ Evaluate $\displaystyle\int_0^1 x^3\,dx$ from the definition using right endpoints and $\sum_{i=1}^n i^3 = \left[\frac{n(n+1)}{2}\right]^2$. Then show your right sum has the form $R_n = \tfrac14 + \tfrac{1}{2n} + \tfrac{1}{4n^2}$, and identify which term is the true integral and which represents the overshoot that decays like $1/n$ (compare the structure of $R_n$ in §13.5).

Part D — Signed Area and Geometry (§13.6–13.7)

D1. ⭐ Evaluate $\displaystyle\int_2^7 5\,dx$ by recognizing a rectangle (§13.7).

D2. ⭐⭐ Evaluate $\displaystyle\int_0^4 (2x + 3)\,dx$ as the area of a trapezoid (§13.7). Show the two parallel side lengths and the width.

D3. ⭐⭐ Evaluate $\displaystyle\int_{-2}^{2} x\,dx$ and explain the cancellation both as two triangles and via the odd-function shortcut (§13.6).

D4. ⭐⭐⭐ Evaluate $\displaystyle\int_{-3}^{3} \sqrt{9 - x^2}\,dx$ by recognizing a semicircle. State the radius and the area formula you use (§13.7).

Part E — Properties of the Definite Integral (§13.8)

E1. ⭐⭐ Given $\displaystyle\int_0^4 f(x)\,dx = 10$ and $\displaystyle\int_0^4 g(x)\,dx = -3$, evaluate $\displaystyle\int_0^4 \big[2f(x) - 5g(x)\big]\,dx$ using linearity.

E2. ⭐⭐ Given $\displaystyle\int_1^5 h\,dx = 12$ and $\displaystyle\int_1^3 h\,dx = 7$, find $\displaystyle\int_3^5 h\,dx$ using additivity over intervals.

E3. ⭐⭐⭐ Use the bounding-box property (§13.8) to show $1 \le \displaystyle\int_0^1 \sqrt{1 + x^3}\,dx \le \sqrt{2}$. Identify $m$ and $M$ explicitly and justify them using the monotonicity of $\sqrt{1+x^3}$ on $[0,1]$.

E4. ⭐⭐⭐⭐ Prove the comparison property from the definition: if $f(x) \le g(x)$ for all $x \in [a,b]$, then $\int_a^b f\,dx \le \int_a^b g\,dx$ (§13.8). (Hint: compare the two Riemann sums term by term for the same sample points and partition, then pass to the limit, citing that limits preserve $\le$.)

Part F — Average Value and the MVT for Integrals (§13.11)

F1. ⭐ Write the formula for the average value $\overline{f}$ of a continuous function on $[a,b]$ (§13.11).

F2. ⭐⭐ Using $\int_0^2 x^2\,dx = \tfrac83$ (from C4), find the average value of $f(x) = x^2$ on $[0, 2]$.

F3. ⭐⭐⭐ For $f(x) = x^2$ on $[0,2]$, the MVT for Integrals guarantees a point $c$ where $f(c) = \overline{f}$ (§13.11). Using $\overline{f}$ from F2, find $c$ explicitly and confirm $c \in [0, 2]$.

Part G — Displacement and Distance (Physics) (§13.9–13.10)

G1. ⭐⭐ A particle moves with constant velocity $v(t) = 6$ m/s for $0 \le t \le 5$ s. Use the integral-as-area interpretation to find its displacement. Then state its total distance and explain why the two agree here (§13.10).

G2. ⭐⭐ A car's velocity (m/s) is recorded every 2 seconds:

Estimate the distance traveled over $[0, 8]$ using the trapezoidal rule (§13.13). Show the weighted sum.

G3. ⭐⭐⭐ A particle has velocity $v(t) = 4 - 2t$ m/s on $[0, 4]$. (a) Using geometry (signed area of triangles), find the net displacement $\int_0^4 v\,dt$. (b) Find the total distance $\int_0^4 |v|\,dt$ by splitting at the sign change of $v$. Explain why the two differ (§13.10).

Part H — Numerical Integration (§13.13)

H1. ⭐⭐ For $\int_0^1 x^2\,dx$ with $n = 2$, compute the midpoint estimate $M_2$ (midpoints $0.25, 0.75$) and the trapezoidal estimate $T_2$. Compare both to the exact value $\tfrac13$ and say which is closer (§13.13).

H2. ⭐⭐⭐ Explain, using the error-scaling table of §13.13, what happens to the trapezoidal-rule error when you double $n$ from 50 to 100 for a smooth integrand. By roughly what factor does the error shrink, and why is that better than the rectangle rules?

H3. ⭐⭐⭐⭐ The following code estimates $\int_0^1 \sqrt{1 - x^2}\,dx$ (a quarter circle, exact value $\pi/4 \approx 0.7854$) by the midpoint rule. The outputs were computed by hand for small $n$. Explain why the midpoint estimate overestimates here, and predict whether refining $n$ removes the bias entirely. (Hint: consider the concavity of $\sqrt{1-x^2}$ and the vertical tangent near $x = 1$ — §13.13 Computational Note.)

import numpy as np

def f(x): return np.sqrt(1 - x**2)

def midpoint(f, a, b, n):
    dx = (b - a) / n
    xm = a + dx * (np.arange(n) + 0.5)
    return dx * np.sum(f(xm))

for n in (4, 20, 100):
    print(n, round(midpoint(f, 0.0, 1.0, n), 5))

# Output (hand-computed):
# 4   0.79716
# 20  0.78809
# 100 0.78569
# True value: pi/4 = 0.78540

Part I — Applied: Accumulation Across Fields (§13.9, §13.14)

I1. ⭐⭐ (Economics) A factory's marginal cost is approximately constant at $C'(q) = 40$ dollars per unit over the production range $q = 100$ to $q = 150$ units. Interpreting added cost as $\int_{100}^{150} C'(q)\,dq$, use the rectangle (constant-integrand) reading to find the added cost of producing units 100 through 150 (§13.9).

I2. ⭐⭐⭐ (Biology) A population grows at a rate $P'(t)$ (individuals/year) sampled annually:

Estimate the net change in population over the 4 years, $\int_0^4 P'(t)\,dt$, using the trapezoidal rule (§13.13). State your answer in individuals.

I3. ⭐⭐⭐ (Data Science / Statistics) The standard normal density is $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ (§13.14). Estimate $P(0 \le X \le 1) = \int_0^1 \phi(x)\,dx$ with a midpoint Riemann sum using $n = 2$ (midpoints $0.25, 0.75$). Use $\phi(0.25) \approx 0.3867$ and $\phi(0.75) \approx 0.3011$. Compare your estimate to the known value $0.3413$ and comment on the accuracy of just two rectangles.

Part J — Synthesis (⭐⭐⭐⭐)

J1. ⭐⭐⭐⭐ For a monotonic function $f$ on $[a,b]$, the right and left Riemann sums bracket the true integral, and their gap is exactly $$R_n - L_n = \big(f(b) - f(a)\big)\,\Delta x = \big(f(b) - f(a)\big)\frac{b-a}{n}$$ (see §13.2). (a) Derive this identity by writing out $R_n - L_n$ as a telescoping difference of the two sums. (b) Use it to bound the error of $L_n$ or $R_n$ for $\int_0^1 x^2\,dx$: how large must $n$ be to guarantee $R_n - L_n < 0.001$? (c) Explain, in terms of this identity, why monotonic functions are the easy case and why a function that oscillates many times poses a harder bracketing problem.

Solutions to odd-numbered problems appear in the back-of-book answer key. Remember: if you reach for an antiderivative, stop — Chapter 14 has not happened yet. Every answer here comes from a sum, a picture, a property, or a numerical rule.