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Free Practice Quiz Question List

10 Applications of Integration Online Quiz Questions

Use this free practice quiz with 20 questions to review 10 Applications of Integration, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

True or false: If two curves cross on an interval, evaluating one signed integral of their difference over the entire interval can underestimate the total geometric area because some contributions may cancel.

  1. A

    True

  2. B

    False

02
Fill in the blank
1 point

Complete the statement: In the general slicing method, the volume of a solid is found by integrating its with respect to the variable.

03
Choose one
1 point

Which definite integral represents the area between f(x)=x+4f(x)=x+4 and g(x)=3−x2g(x)=3-\frac{x}{2} from x=1x=1 to x=4x=4?

  1. A

    ∫14[(3−x2)−(x+4)]dx\int_1^4\left[(3-\frac{x}{2})-(x+4)\right]dx

  2. B

    ∫14[(x+4)−(3−x2)]dx\int_1^4\left[(x+4)-(3-\frac{x}{2})\right]dx

  3. C

    ∫14[(x+4)+(3−x2)]dx\int_1^4\left[(x+4)+(3-\frac{x}{2})\right]dx

  4. D

    ∫14[(x+4)(3−x2)]dx\int_1^4\left[(x+4)(3-\frac{x}{2})\right]dx

04
Written response
1 point

A spring follows Hooke's law F(x)=4xF(x)=4x newtons. What work, in joules, is required to stretch it from x=1x=1 metre to x=3x=3 metres?

05
True or false
1 point

True or false: The average value of a continuous function on an interval must equal the function's value at the midpoint of that interval.

  1. A

    True

  2. B

    False

06
Fill in the blank
1 point

For a disk whose density depends on the distance rr from its center, the mass of a thin ring is modeled using dm=dm=ρ(r) dr\rho(r)\,dr.

07
Choose one
1 point

The region between y=xy=\sqrt{x} and y=1y=1, for 1≤x≤41\le x\le4, is revolved about the x-axis. Which integral gives the resulting volume?

  1. A

    π∫14x dx\pi\int_1^4 x\,dx

  2. B

    2π∫14xx dx2\pi\int_1^4 x\sqrt{x}\,dx

  3. C

    π∫14(x−1) dx\pi\int_1^4(x-1)\,dx

  4. D

    π∫14(1−x) dx\pi\int_1^4(1-x)\,dx

08
Choose all
1 point

Select all statements that correctly describe cylindrical shells formed by revolving vertical rectangles about the y-axis.

  1. A

    For rotation about the y-axis, a vertical shell at x has radius x.

  2. B

    The shell height is the vertical length of the representative rectangle.

  3. C

    The shell radius is always the function value f(x)f(x).

  4. D

    The shell volume is modeled by 2π∫(radius)(height) dx2\pi\int (\text{radius})(\text{height})\,dx.

09
Written response
1 point

Find the average value of f(x)=2x+1f(x)=2x+1 on the interval [1,4][1,4].

10
Choose one
1 point

A vertical rectangular plate has width 2 and extends from y=0y=0 to y=3y=3, with the liquid surface at y=3y=3. If the liquid's weight density is γ\gamma, which integral gives the hydrostatic force on the plate?

  1. A

    ∫03γy(2) dy\int_0^3\gamma y(2)\,dy

  2. B

    ∫03γ(3−y)(2) dy\int_0^3\gamma(3-y)(2)\,dy

  3. C

    ∫03γ(3+y)(2) dy\int_0^3\gamma(3+y)(2)\,dy

  4. D

    ∫03γ(3−y)2(2) dy\int_0^3\gamma(3-y)^2(2)\,dy

11
Choose all
1 point

Select all statements that correctly model the work required to pump a liquid to a destination height.

  1. A

    The lifting distance is measured from the layer's height to the destination height.

  2. B

    A layer's work contribution can be written as dW=γA(y)D(y) dydW=\gamma A(y)D(y)\,dy.

  3. C

    The lifting distance is always equal to the layer's thickness.

  4. D

    The total work is obtained by integrating the contributions of all layers.

12
Choose one
1 point

If a bounded region has right boundary x=R(y)x=R(y) and left boundary x=L(y)x=L(y) for c≤y≤dc\le y\le d, which integral represents its area?

  1. A

    ∫cd[R(y)−L(y)] dy\int_c^d[R(y)-L(y)]\,dy

  2. B

    ∫cd[L(y)−R(y)] dy\int_c^d[L(y)-R(y)]\,dy

  3. C

    ∫cd[R(y)+L(y)] dy\int_c^d[R(y)+L(y)]\,dy

  4. D

    ∫cdR(y)L(y) dy\int_c^d R(y)L(y)\,dy

13
Open ended
1 point

Explain how to set up the volume when the region under y=f(x)y=f(x), above the x-axis, from x=ax=a to x=bx=b, is revolved about the y-axis. Identify the shell dimensions, write the volume integral, and explain why cylindrical shells may be preferable to washers.

14
Choose one
1 point

A solid extends from x = 0 to x = 2, and its cross-sectional area perpendicular to the x-axis is A(x) = x² + 1. Which definite integral gives the volume of the solid?

  1. A

    V = ∫₀² (x² + 1)² dx

  2. B

    V = ∫₀² (x² + 1) dx

  3. C

    V = π∫₀² (x² + 1) dx

  4. D

    V = ∫₀² x² dx + 1

15
Choose one
1 point

Which integral represents the area bounded above by y = x + 4, below by y = 3 − x/2, and between x = 1 and x = 4?

  1. A

    ∫₁⁴ [(3 − x/2) − (x + 4)] dx

  2. B

    ∫₁⁴ [(x + 4) + (3 − x/2)] dx

  3. C

    ∫₁⁴ [(x + 4) − (3 − x/2)] dx

  4. D

    ∫₁⁴ [(x + 4) − (3 − x/2)] dy

16
True or false
1 point

For a pumping problem, the lifting distance for each thin liquid layer should be measured from that layer’s height to the destination height.

  1. A

    True

  2. B

    False

17
Written response
1 point

Find the average value of f(x)=x2f(x)=x^2 on the interval [0,3][0,3]. Enter the average value as a number. The absolute tolerance is 0.

18
Choose one
1 point

The region between y = √x and y = 1 for 1 ≤ x ≤ 4 is revolved around the x-axis. Which integral gives the resulting volume?

  1. A

    π∫₁⁴ (x − 1) dx

  2. B

    π∫₁⁴ (x + 1) dx

  3. C

    π∫₁⁴ (√x − 1) dx

  4. D

    2π∫₁⁴ x dx

19
Written response
1 point

A spring has force F(x)=4xF(x)=4x. How much work is required to stretch it from x=1x=1 to x=3x=3? Enter the work as a number. The absolute tolerance is 0.

20
Choose one
1 point

A disk has radius 2, and its density at distance r from the center is ρ(r) = r. Which integral gives its mass?

  1. A

    m = ∫₀² 2πr² dr

  2. B

    m = ∫₀² 2πr dr

  3. C

    m = ∫₀² πr² dr

  4. D

    m = ∫₀² 2πr · 2 dr