Q: If $f(x)$ is continuous on $[a, b]$, which of the following statements is true about $\int_a^b f(x)\,dx$?

It can be computed as the limit of Riemann sums as $n \to \infty$.

Q: Which of the following is the correct limit definition of the definite integral $\int_a^b f(x)\,dx$?

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

Question 1

Which of the following best describes the meaning of a definite integral $\int_a^b f(x)\,dx$?

Accepted Answer

It represents the total accumulation of $f(x)$ over $[a, b]$, often interpreted as the area under the curve $y=f(x)$ from $x=a$ to $x=b$.

Question 2

Suppose you want to estimate the area under $y=f(x)$ from $x=a$ to $x=b$ using $n$ rectangles of equal width. What is the width $\Delta x$ of each rectangle?

Accepted Answer

$\Delta x = \frac{b-a}{n}$

Question 3

Which of the following expressions represents a right-endpoint Riemann sum for $f(x)$ on $[a, b]$ with $n$ subintervals?

Accepted Answer

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

Question 4

The Fundamental Theorem of Calculus states that if $F$ is an antiderivative of $f$ on $[a, b]$, then:

Accepted Answer

$\int_a^b f(x)\,dx = F(b) - F(a)$

Question 5

Which of the following is the correct setup for the area between the curves $y=f(x)$ and $y=g(x)$ on $[a, b]$, where $f(x) \geq g(x)$?

Accepted Answer

$\int_a^b [f(x) - g(x)]\,dx$

Question 6

A biologist measures the rate at which a chemical is produced in a reaction as $r(t)$ mg/min over $0 \leq t \leq 10$ minutes. What does $\int_0^{10} r(t)\,dt$ represent?

Accepted Answer

The total amount of chemical produced in 10 minutes.

Question 7

Given $\int_1^4 (2x+1)\,dx$, what is the value of the definite integral?

Accepted Answer

15

Question 8

If $f(x)$ is continuous on $[a, b]$, which of the following statements is true about $\int_a^b f(x)\,dx$?

Accepted Answer

It can be computed as the limit of Riemann sums as $n 	o \infty$.

Question 9

A physics student wants to find the work done by a variable force $F(x)$ moving an object from $x=a$ to $x=b$. Which integral should they use?

Accepted Answer

$\int_a^b F(x)\,dx$

Question 10

Which of the following is a correct interpretation of $\int_0^2 x^2\,dx$?

Accepted Answer

The area under $y=x^2$ from $x=0$ to $x=2$.

Question 11

If $\int_1^3 f(x)\,dx = 5$ and $\int_3^5 f(x)\,dx = 7$, what is $\int_1^5 f(x)\,dx$?

Accepted Answer

12

Question 12

Which of the following is the correct setup for the area between $y=\sin x$ and $y=1-\sin x$ on $[0, \frac{\pi}{2}]$?

Accepted Answer

$\int_0^{\frac{\pi}{2}} [\sin x - (1-\sin x)]\,dx$

Question 13

A region is bounded by $y=x^2$, $y=0$, and $x=2$. What is the area of this region?

Accepted Answer

$\int_0^2 x^2\,dx$

Question 14

Which of the following is NOT a possible application of definite integrals as described in the study materials?

Accepted Answer

Determining the slope of a tangent line.

Question 15

If $f(x)$ is negative on $[a, b]$, what does $\int_a^b f(x)\,dx$ represent?

Accepted Answer

A negative value, representing the net area below the $x$-axis.

Question 16

Which of the following is the correct limit definition of the definite integral $\int_a^b f(x)\,dx$?

Accepted Answer

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

Question 17

If $\int_0^3 f(x)\,dx = 8$ and $\int_3^5 f(x)\,dx = -2$, what is $\int_0^5 f(x)\,dx$?

Accepted Answer

6

Question 18

Which of the following steps is NOT part of the process for finding the area between two curves?

Accepted Answer

Differentiate both functions.

Applications of Integration: Area and the Definite Integral

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