Chapter 13 — Quiz
10 questions covering the area problem, Riemann sums, the integral as a limit, signed area, properties, average value, and numerical integration. Answers and the relevant section appear under each. No question requires the Fundamental Theorem (that is Chapter 14).
Work each problem before opening the answer. Aim to justify your reasoning, not just match a number.
1. For $f(x) = x^2$ on $[0, 2]$ with $n = 4$ equal subintervals, what is $\Delta x$, and what are the right endpoints?
Answer
$\Delta x = \dfrac{2 - 0}{4} = 0.5$. The right endpoints are $x_i = 0 + i(0.5)$ for $i = 1,2,3,4$, namely $0.5,\ 1.0,\ 1.5,\ 2.0$. (§13.3)2. Evaluate $\displaystyle\sum_{i=1}^{4} (2i^2 - 1)$.
Answer
Write out the terms: $i=1\!: 2(1)-1 = 1$; $i=2\!: 2(4)-1 = 7$; $i=3\!: 2(9)-1=17$; $i=4\!: 2(16)-1 = 31$. Sum $= 1 + 7 + 17 + 31 = 56$. (Check via formulas: $2\sum i^2 - \sum 1 = 2(30) - 4 = 56$.) (§13.3)3. A function $f$ is increasing on $[a,b]$. Which is larger for the same regular partition, the left Riemann sum $L_n$ or the right Riemann sum $R_n$? Which one overestimates the true integral?
Answer
For an increasing function, each subinterval's largest value is at its **right** endpoint, so $R_n > L_n$, and $R_n$ **overestimates** while $L_n$ underestimates the true area. (For a decreasing function the roles swap.) (§13.2)4. Evaluate $\displaystyle\int_0^1 x\,dx$ from the definition using right endpoints and $\sum_{i=1}^n i = \frac{n(n+1)}{2}$.
Answer
$\Delta x = 1/n$, $x_i = i/n$, so $$R_n = \sum_{i=1}^n \frac{i}{n}\cdot\frac1n = \frac{1}{n^2}\cdot\frac{n(n+1)}{2} = \frac{n+1}{2n} = \frac12 + \frac{1}{2n}.$$ As $n \to \infty$, $R_n \to \tfrac12$. So $\int_0^1 x\,dx = \tfrac12$ — the area of a triangle with base 1, height 1. (§13.5)5. What does the definite integral $\int_a^b f(x)\,dx$ measure when $f$ takes both positive and negative values on $[a,b]$? State the formula in words.
Answer
It measures **signed area**: $\int_a^b f\,dx = (\text{area above the axis}) - (\text{area below the axis})$. Regions where $f < 0$ contribute negatively. For total (unsigned) area between the graph and the axis, integrate $|f|$ instead. (§13.6)6. Evaluate $\displaystyle\int_0^3 (2x + 1)\,dx$ using geometry (no antiderivatives).
Answer
The region under $y = 2x + 1$ over $[0,3]$ is a trapezoid with parallel vertical sides $f(0) = 1$ and $f(3) = 7$ and width $3$. Area $= \dfrac{1 + 7}{2}\cdot 3 = 4 \cdot 3 = 12$. (§13.7)7. Given $\displaystyle\int_0^6 f\,dx = 14$ and $\displaystyle\int_0^2 f\,dx = 5$, find $\displaystyle\int_2^6 f\,dx$. Which property do you use?
Answer
By **additivity over intervals** (§13.8), $\int_0^2 f + \int_2^6 f = \int_0^6 f$, so $\int_2^6 f = 14 - 5 = 9$. (§13.8)8. Find the average value of $f(x) = x^2$ on $[0, 1]$. You may use $\int_0^1 x^2\,dx = \tfrac13$.
Answer
$\overline{f} = \dfrac{1}{1 - 0}\int_0^1 x^2\,dx = \dfrac{1}{1}\cdot\dfrac13 = \dfrac13$. By the MVT for Integrals there is a $c \in [0,1]$ with $f(c) = \tfrac13$, namely $c = 1/\sqrt3 \approx 0.577$. (§13.11)9. For $\int_0^1 x^2\,dx$, compute the midpoint estimate $M_2$ with $n = 2$ (midpoints $0.25$ and $0.75$). How close is it to the exact $\tfrac13$?
Answer
$\Delta x = 0.5$. Heights: $f(0.25) = 0.0625$, $f(0.75) = 0.5625$. So $$M_2 = 0.5(0.0625 + 0.5625) = 0.5(0.625) = 0.3125.$$ The exact value is $0.3333\ldots$, so $M_2$ is off by about $0.021$ (roughly $6\%$) — already close with only two rectangles, and far better than the endpoint rules at this $n$. (§13.13)10. The standard normal density is $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Why can't we evaluate $\int_a^b \phi(x)\,dx$ with an elementary antiderivative, and how do statistical tables get their numbers?
Answer
$\phi$ has **no elementary antiderivative** — there is no formula built from polynomials, roots, exponentials, logarithms, and trig functions whose derivative is $\phi$ (a fact proved by Liouville). Statistical tables, $z$-scores, and the "68–95–99.7" rule are computed by **numerical integration** — Riemann-type sums or the trapezoidal rule of §13.13 applied to $\phi$. (§13.14)Scoring Guide
| Score | Interpretation |
|---|---|
| 9–10 | Excellent. You can build the definite integral from Riemann sums, read signed area, apply the properties, and integrate numerically. Ready for the Fundamental Theorem in Chapter 14. |
| 7–8 | Solid. Review any missed item — most likely sigma-notation bookkeeping (§13.3) or the signed-area vs. total-area distinction (§13.6). |
| 5–6 | Partial. Re-read §13.3–13.5 (sums and the limit definition) and §13.8 (properties), then redo questions 4, 7, and 9. |
| 0–4 | Revisit the chapter from §13.2. Focus on the slice–sum–limit pattern and work the Check Your Understanding boxes before retrying. |
If you found yourself wanting an antiderivative, that instinct is correct — but it belongs to Chapter 14. Everything here is sums, pictures, and limits.