Free Practice Quiz Question List

Applications of Integration Online Quiz Questions

Use this free practice quiz with 20 questions to review Applications of Integration, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

Two continuous curves satisfy $f(x)\ge g(x)$ on $[a,b]$ . Which integral gives the area between the curves using vertical slices?

A
$\int_a^b [g(x)-f(x)]\,dx$
B
$\int_a^b [f(x)-g(x)]\,dx$
C
$\int_a^b [f(x)+g(x)]\,dx$
D
$\int_a^b |f(x)+g(x)|\,dx$

Choose one

1 point

A solid has washer cross-sections with outer radius $R(x)$ and inner radius $r(x)$ . What is the cross-sectional area?

A
$\pi(R^2-r^2)$
B
$\pi(R-r)^2$
C
$2\pi(R-r)$
D
$\pi(R^2+r^2)$

Choose one

1 point

Which expression represents the average value of a continuous function $f$ on $[a,b]$ ?

A
$\frac{f(b)-f(a)}{b-a}$
B
$\int_a^b f(x)\,dx$
C
$\frac{1}{b-a}\int_a^b f(x)\,dx$
D
$f(b)-f(a)$

Choose all

1 point

Select all statements that correctly distinguish displacement from total distance for one-dimensional motion.

A
Displacement can be negative.
B
Total distance is found by integrating speed.
C
Displacement is always greater than total distance.
D
Velocity must be considered on intervals separated by its zeros when finding total distance.

Choose all

1 point

Select all statements that correctly describe vertical cylindrical shells formed by rotating a region about the y-axis.

A
The shell radius is the distance from the y-axis.
B
The shell circumference is $2\pi$ times the radius.
C
The shell height is the sum of the boundary functions.
D
For a region between $f$ and $g$ , the shell height is $f(x)-g(x)$ when $f\ge g$ .

True or false

1 point

If $A(x)=\int_a^x r(t)\,dt$ , then $A'(x)=r(x)$ .

A
True
B
False

True or false

1 point

For two curves that cross inside $[a,b]$ , a single integral of one fixed function minus the other always gives the enclosed area without splitting the interval.

A
True
B
False

Written response

1 point

For $f(x)=x^2$ on $[0,3]$ , calculate the average value of $f$ . Enter the exact numerical value.

Fill in the blank

1 point

For a solid whose cross-sectional area perpendicular to the x-axis is $A(x)$ , the volume is found by integrating the with respect to x.

Fill in the blank

1 point

To calculate total distance traveled over $[a,b]$ , integrate the with respect to time.

Open ended

1 point

The region under $y=x$ from $x=0$ to $x=2$ is revolved about the y-axis. Set up and evaluate a cylindrical-shell integral for the volume, explaining the radius and height of a representative shell.

Choose one

1 point

Let $A(x)=\int_a^x r(t)\,dt$ . Which expression gives the derivative of the accumulation function?

A
$A'(x)=\int_a^x r(t)\,dt$
B
$A'(x)=r(x)$
C
$A'(x)=r'(x)$
D
$A'(x)=a\,r(x)$

Choose one

1 point

What is the area enclosed by $y=2x$ , $y=x^2$ , and the lines $x=0$ and $x=2$ ?

A
$\frac{2}{3}$
B
$\frac{4}{3}$
C
$2$
D
$\frac{8}{3}$

Choose one

1 point

The region under $y=\sqrt{x}$ and above the $x$ -axis for $0\le x\le4$ is revolved about the $x$ -axis. Which integral represents the resulting volume?

A
$\pi\int_0^4\sqrt{x}\,dx$
B
$2\pi\int_0^4x\sqrt{x}\,dx$
C
$\pi\int_0^4x\,dx$
D
$\pi\int_0^4(4-x)\,dx$

Choose one

1 point

The region under $y=3-x$ , above the $x$ -axis, and between $x=0$ and $x=3$ is revolved about the $y$ -axis. What is the volume?

A
$\frac{9\pi}{2}$
B
$6\pi$
C
$8\pi$
D
$9\pi$

Choose one

1 point

What is the average value of $f(x)=x^2$ on the interval $[1,3]$ ?

A
$\frac{13}{3}$
B
$\frac{14}{3}$
C
$\frac{26}{3}$
D
$\frac{9}{2}$

True or false

1 point

True or false: Over an interval of motion, total distance traveled is always equal to $\int_a^b v(t)\,dt$ .

A
True
B
False

Written response

1 point

A particle has velocity $v(t)=2t-3$ for $0\le t\le3$ . Enter its displacement over this interval as an exact number of distance units.

Written response

1 point

A solid has square cross-sections perpendicular to the $x$ -axis, with side length $s(x)=1+x$ for $0\le x\le2$ . Enter the volume as an exact number of cubic units.

Written response

1 point

A quantity has initial value Q(0)=5 units and rate of change Q′(t)=4t−1 units per minute. What is Q(2)? Enter the final amount in units.