Question 1

Which of the following best represents the general Riemann sum for approximating the area under the curve $f(x)$ on $[a, b]$?

Accepted Answer

$\sum_{i=1}^{n} f(x_i^*) \Delta x$

Question 2

Given $\int_{a}^{b} f(x) dx = F(b) - F(a)$, which theorem is being applied?

Accepted Answer

Fundamental Theorem of Calculus

Question 3

Evaluate $\int \sin(2x) dx$.

Accepted Answer

$-\frac{1}{2} \cos(2x) + C$

Question 4

Which of the following is always true for a continuous function $f$ on $[a, b]$ if $f(x) \geq 0$ for all $x$ in $[a, b]$?

Accepted Answer

$\int_{a}^{b} f(x) dx \geq 0$

Question 5

Suppose $f(x)$ is continuous and increasing on $[a, b]$. Which Riemann sum method will always overestimate the definite integral?

Accepted Answer

Right-hand sum

Question 6

Find $\int \frac{1}{1 + (2x)^2} dx$.

Accepted Answer

$\frac{1}{2} \arctan(2x) + C$

Question 7

If $F(x) = \int_{1}^{x^2} e^{t^2} dt$, what is $F'(x)$?

Accepted Answer

$2x e^{x^4}$

Question 8

Which of the following is the correct formula for the average value of a continuous function $f$ on $[a, b]$?

Accepted Answer

$\frac{1}{b-a} \int_{a}^{b} f(x) dx$

Question 9

Evaluate $\int \sec^2(x) dx$.

Accepted Answer

$	an(x) + C$

Question 10

A particle moves with velocity $v(t) = 5t^2 + 2t - 6$. What is the total distance traveled from $t = 1$ to $t = 3$?

Accepted Answer

$\int_{1}^{3} (5t^2 + 2t - 6) dt$

Question 11

Which property of definite integrals is illustrated by $\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx$?

Accepted Answer

Reversal of limits

Question 12

Evaluate $\int \frac{1}{\sqrt{1 - (2x)^2}} dx$.

Accepted Answer

$\frac{1}{2} \arcsin(2x) + C$

Question 13

Which of the following is the correct expression for the midpoint Riemann sum for $f(x)$ on $[a, b]$ with $n$ subintervals?

Accepted Answer

$\sum_{i=1}^{n} f(x_i^*) \Delta x$

Question 14

If $f(x)$ is continuous on $[a, b]$ and $f(x) \leq g(x)$ for all $x$ in $[a, b]$, which of the following is always true?

Accepted Answer

$\int_{a}^{b} f(x) dx \leq \int_{a}^{b} g(x) dx$

Question 15

Evaluate $\int x^n dx$ for $n 
eq -1$.

Accepted Answer

$\frac{x^{n+1}}{n+1} + C$

Question 16

Which of the following is the correct formula for the definite integral of $f(x)$ from $a$ to $b$ using the Fundamental Theorem of Calculus?

Accepted Answer

$F(b) - F(a)$, where $F$ is any antiderivative of $f$

Question 17

Evaluate $\int \sec(x) dx$.

Accepted Answer

$\ln|\sec(x) + 	an(x)| + C$

Question 18

If $f(x)$ is continuous on $[a, b]$ and $f(x) < 0$ for all $x$ in $[a, b]$, which of the following is always true?

Accepted Answer

$\int_{a}^{b} f(x) dx < 0$

Question 19

Which of the following is the correct formula for the sum $\sum_{k=1}^{n} k^2$?

Accepted Answer

$\frac{n(n+1)(2n+1)}{6}$

Calculus II: Definite Integrals, Riemann Sums, and Applications – Midterm #1 Review

Study Guide - Practice Questions