Free Practice Quiz Question List

2 Applications of Integration Online Quiz Questions

Use this free practice quiz with 20 questions to review 2 Applications of Integration, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: If two continuous curves are described as functions of x and one remains above the other on [a,b], the area between them is found by integrating the upper function minus the lower function.

A
True
B
False

Choose one

1 point

Which expression gives the average value of $f(x)=x^2$ on the interval $[0,3]$ ?

A
$\int_0^3 x^2\,dx$
B
$\frac{1}{3}\int_0^3 x^2\,dx$
C
$3\int_0^3 x^2\,dx$
D
$\frac{1}{2}\int_0^3 x^2\,dx$

Written response

1 point

A region is revolved around the y-axis. Vertical slices give a simple radius and height, while horizontal slices would require solving for x as a function of y. What integration method is most appropriate?

Fill in the blank

1 point

For a force $F(x)$ that varies with position from $x=a$ to $x=b$ , the work is $W=$ .

Choose one

1 point

Which integral gives the arc length of the curve $y=f(x)$ from $x=a$ to $x=b$ , assuming $f'$ is continuous?

A
$\int_a^b f'(x)\,dx$
B
$\int_a^b\sqrt{1-f'(x)}\,dx$
C
$\int_a^b\sqrt{1+[f'(x)]^2}\,dx$
D
$\int_a^b [f'(x)]^2\,dx$

True or false

1 point

True or false: In a fluid with constant weight density, a horizontal strip experiences greater pressure when it is located deeper below the fluid surface.

A
True
B
False

Choose all

1 point

Select all statements that correctly describe methods for finding volumes of solids.

A
The volume of a solid with cross-sectional area A(x) perpendicular to the x-axis is $\int_a^b A(x)\,dx$ .
B
For washers, the cross-sectional area is always $\pi(R+r)^2$ .
C
A cylindrical shell has approximate volume equal to circumference times height times thickness.
D
For rotation about an axis, a radius may be used as a signed coordinate even when it is not a distance.

Written response

1 point

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

Fill in the blank

1 point

For a thin rod on $[a,b]$ with linear density $\rho(x)$ , the mass is $m=$ . For a planar object, its center-of-mass x-coordinate is $\bar{x}=$ .

Choose one

1 point

The rectangle $0\le x\le 2$ , $1\le y\le 3$ is revolved about the x-axis. What is the resulting volume?

A
$2\pi$
B
$16\pi$
C
$8\pi$
D
$24\pi$

Choose all

1 point

Select all statements that are correct about spring work and lifting or pumping work.

A
For a spring obeying Hooke's law, the force required at displacement x is $F(x)=kx$ .
B
The work required to move a spring from x=a to x=b is always $k(b-a)$ .
C
The work for a spring moved from x=a to x=b is $\frac{k}{2}(b^2-a^2)$ .
D
In a pumping problem, lifting distance can be replaced by the layer's vertical coordinate without reference to the destination.

Choose one

1 point

Which integral gives the surface area generated by revolving $y=f(x)$ about the y-axis from $x=a$ to $x=b$ , when x is the nonnegative distance to the axis?

A
$2\pi\int_a^b f(x)\,dx$
B
$2\pi\int_a^b x\sqrt{1+[f'(x)]^2}\,dx$
C
$\pi\int_a^b x^2[ f'(x)]^2\,dx$
D
$2\pi\int_a^b f'(x)\sqrt{1+x^2}\,dx$

Open ended

1 point

Find the area enclosed by $y=x$ and $y=x^2$ . Explain how you determine the bounds and which function is the upper curve.

Choose one

1 point

Which definite integral represents the area enclosed by $y=x$ , $y=x^2$ , $x=0$ , and $x=1$ ?

A
$\int_0^1(x^2-x)\,dx$
B
$\int_0^1(x-x^2)\,dx$
C
$\int_0^1(x+x^2)\,dx$
D
$\int_0^1\lvert x-x^2\rvert^2\,dx$

Choose one

1 point

A solid has cross-sectional area $A(x)$ perpendicular to the $x$ -axis for $a\le x\le b$ . Which expression gives its volume?

A
$V=\int_a^b xA(x)\,dx$
B
$V=\pi\int_a^b A(x)^2\,dx$
C
$V=\int_a^b A(x)\,dx$
D
$V=\int_a^b A'(x)\,dx$

Choose one

1 point

Which formula gives the arc length of $y=f(x)$ from $x=a$ to $x=b$ , assuming $f'$ is continuous?

A
$\int_a^b\sqrt{1+[f'(x)]^2}\,dx$
B
$\int_a^b\sqrt{1+f(x)^2}\,dx$
C
$\int_a^b f'(x)\,dx$
D
$\int_a^b[1+f'(x)]^2\,dx$

Choose one

1 point

Which expression represents the average value of $f(x)=x^2$ on $[0,3]$ ?

A
$\frac{1}{3}\int_0^3x\,dx$
B
$\int_0^3x^2\,dx$
C
$\frac{1}{2}\int_0^3x^2\,dx$
D
$\frac{1}{3}\int_0^3x^2\,dx$

True or false

1 point

True or false: In a hydrostatic-force calculation, the pressure at a point is determined by its depth below the fluid surface, not merely by its vertical coordinate.

A
True
B
False

Written response

1 point

What is the standard term for the parameter $k$ in Hooke's law $F(x)=kx$ ?

Written response

1 point

A spring has constant $k=4$ and is stretched from $x=1$ metre to $x=3$ metres. What work is required? Enter the exact value in joules.