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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 8: Applications of IntegrationVolumes of Revolution

Volumes of Revolution (College Board AP® Calculus AB): Exam Questions

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Volumes of Revolution

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1
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A solid metal sculpture has the shape obtained by revolving the curve y equals 3 plus cos space 2 x from x equals 0 to x equals 4 about the x-axis, where x and y are measured in inches. What volume of metal, in cubic inches, was used to create the sculpture?

  • 12.495

  • 39.253

  • 40.932

  • 128.592

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2
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Graph of the curve y = 4sin(x) and the lines y = 4 and x = pi/3. The area bounded by the two lines, the curve and the y-axis is shaded and labeled R.

Let R be the region enclosed by the line y equals 4, the line x equals pi over 3, and the curve y equals 4 space sin space x, as shown in the figure above. A solid is generated by rotating R about the line y equals 4.

Which of the following definite integrals gives the volume of the solid?

  • pi integral subscript 0 superscript pi over 3 end superscript open parentheses 4 minus 4 space sin space x close parentheses space d x

  • pi integral subscript 0 superscript pi over 3 end superscript open parentheses 4 minus 4 space sin space x close parentheses squared space d x

  • pi integral subscript 0 superscript 4 open parentheses 4 minus 4 space sin space x close parentheses space d x

  • pi integral subscript 0 superscript 4 open parentheses 4 minus 4 space sin space x close parentheses squared space d x

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3
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Graph showing a shaded region labeled R. The region R is bounded by the curve y = e^x, the line y = 3 and the y-axis.

Let R be the region bounded above by the line y equals 3, to the right by the y equals e to the power of x, and to the left by the y-axis as shown in the figure above.

A solid is generated by revolving R around the y-axis. Which of the following definite integrals gives the volume of the solid?

  • pi integral subscript 1 superscript 3 e to the power of 2 y end exponent space d y

  • pi integral subscript 1 superscript 3 ln squared y space d y

  • pi integral subscript 0 superscript ln space 3 end superscript e to the power of 2 y end exponent space d y

  • pi integral subscript 0 superscript ln space 3 end superscript ln squared y space d y

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4
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Graph showing a shaded region labeled R. The shaded region R is bounded above by the curve y = x^2 + 2, to the left by the y-axis, to the right by a vertical line at x = 2 and below by the horizontal line y=1.

Let R be the region enclosed by the curve y equals x squared plus 2, the y-axis and the lines y equals 1 and x equals 2 as shown in the figure above.

A solid is generated by revolving R about the x-axis. What is the volume of the solid?

  • 23.067

  • 43.145

  • 72.466

  • 78.749

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5
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Graph showing a shaded region labeled R. The shaded region R is bounded below by the curve y = x^2, to the right by the vertical line x = -1 and above by the horizontal line y = 9.

Let R be the region enclosed by the curve y equals x squared, the lines y equals 9 and x equals negative 1 as shown in the figure above.

A solid is generated by revolving R about the y-axis. Which of the following definite integrals gives the volume of the solid?

  • pi integral subscript 0 superscript 9 y minus 1 space d y

  • pi integral subscript 1 superscript 9 y minus 1 space d y

  • pi integral subscript negative 3 end subscript superscript 0 81 minus x to the power of 4 space d x

  • pi integral subscript negative 3 end subscript superscript negative 1 end superscript 81 minus x to the power of 4 space d x

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1
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A region in the first quadrant is enclosed by the graphs of y equals x cubed and y equals square root of x. If the region is rotated about the x-axis, which of the following represents the volume of the solid that is generated?

  • table row blank blank cell integral subscript 0 superscript 1 open parentheses square root of x minus x cubed close parentheses space d x end cell end table

  • table row blank blank cell integral subscript 0 superscript 1 pi open parentheses square root of x minus x cubed close parentheses squared space d x end cell end table

  • table row blank blank cell integral subscript 0 superscript 1 pi x open parentheses 1 minus x to the power of 5 close parentheses space d x end cell end table

  • table row blank blank cell integral subscript 0 superscript 1 pi x to the power of 7 over 2 end exponent space d x end cell end table

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2
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Let R be the region in the first quadrant enclosed by the coordinate axes, the line y equals 4, and the curve with equation y equals x squared minus 3, as shown in the figure below.

A graph showing the shaded region R in the first quadrant, bounded by the coordinate axes, the line y=4 and the curve y=x^2-3

What is the volume of the solid generated when R is rotated about the y-axis?

  • 2 over 3 open parentheses 7 square root of 7 minus 3 square root of 3 close parentheses pi

  • fraction numerator 28 pi over denominator 3 end fraction

  • 20 pi

  • 100 over 3 pi y squared plus 3

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3
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What is the volume of the solid generated when the region bounded by the graph of x equals square root of y minus 1 end root and the lines x equals 0 and y equals 7 is revolved about the y-axis?

  • 18.000

  • 30.781

  • 54.978

  • 56.549

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4
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Let R be the region bounded by the graphs of y equals 2 to the power of x, y equals 2 to the power of 4, and x equals 0. Which of the following finds the volume of the solid formed by revolving R about the line y equals negative 3?

  • pi integral subscript 0 superscript 4 space open parentheses 2 to the power of 4 plus 3 close parentheses squared minus open parentheses 2 to the power of x plus 3 close parentheses squared space d x

  • pi integral subscript 0 superscript 4 space open parentheses 2 to the power of 4 minus 2 to the power of x minus 3 close parentheses squared space d x

  • pi integral subscript 0 superscript 4 space open parentheses 2 to the power of 4 minus 2 to the power of x plus 3 close parentheses squared space d x

  • pi integral subscript 0 superscript 4 space open parentheses 2 to the power of 4 minus 2 to the power of x close parentheses squared plus 3 space d x

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5
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Graphs of y = x/5 and y = 6 - x. The point of intersection of the two straight lines is labeled (5, 1) and the intersections of the straight lines with the x-axis occur at x = 0 and x = 6. The region bound by the two straight lines and the x-axis is shaded and labeled R.

Let R be the region enclosed by the line y equals x over 5, the line y equals 6 minus x, and the x-axis as shown in the figure above.

What is the volume of the solid generated by revolving R about the line x equals 6.

  • 14 pi

  • 22 pi

  • 430 pi

  • 864 pi

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1
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A graph of the shaded region enclosed by the ellipse with equation x^2 + 4y^2 = 16, stretching from x=-4 to x=4 along the x-axis, and y=-2 to y=2 along the y-axis

The shaded region shown in the figure above is the region enclosed by the curve with equation x squared plus 4 y squared equals 16. The region is symmetrical about both the x- and y-axes. The volume of the solid obtained by revolving the region about the x-axis is

  • fraction numerator 32 pi over denominator 3 end fraction

  • fraction numerator 64 pi over denominator 3 end fraction

  • 32 pi

  • 48 pi

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2
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Let R be the region enclosed by the graph of f open parentheses x close parentheses equals arcsin x, the line y equals pi over 2, and the y-axis, as shown in the figure below.

A graph showing the shaded region R enclosed by the curve y=arcsinx, the line y=pi/2, and the y=-axis

What is the volume of the solid generated when R is rotated about the vertical line line x equals 1?

  • 1.119

  • 1.215

  • 2.467

  • 3.816

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3
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A region in the first quadrant is enclosed by the graphs of y equals e to the power of 3 x end exponent, x equals 1, and the coordinate axes. If the region is rotated about the y-axis, which of the following represents the volume of the solid that is generated?

  • 2 pi integral subscript 0 superscript 1 x e to the power of 3 x end exponent space d x

  • pi integral subscript 0 superscript 1 e to the power of 6 x end exponent space d x

  • table row blank blank cell pi over 9 end cell end table integral subscript 1 superscript e cubed end superscript table row blank blank ln end table squared table row blank blank y end table table row blank blank space end table table row blank blank d end table table row blank blank y end table

  • pi e cubed minus table row blank blank cell pi over 9 end cell end table integral subscript 1 superscript e cubed end superscript table row blank blank ln end table squared table row blank blank y end table table row blank blank space end table table row blank blank d end table table row blank blank y end table

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4
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Graph of function y = sin x, which intersects the x-axis at x = -pi and x = 0. The region bounded above by the x-axis and below by y = sin x, in the third quadrant, is shaded and labeled R. The line y = -1 is also drawn on the graph and labeled.

Let R be the region in the third quadrant bounded by the x-axis and the curve y equals sin space x as shown in the diagram above.

A solid is generated by revolving R about the line y equals negative 1. What is the volume of the solid?

  • 2.032

  • 2.238

  • 7.632

  • 12.108

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5
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Graph showing a shaded region on a graph labeled R. R is bounded by the curve y = squareroot(x) and the straight line y = (1/7)(x + 10).

The region R is enclosed by the curve y equals square root of x, and the line y equals 1 over 7 open parentheses x plus 10 close parentheses as shown in the figure above.

What is the volume of the solid generated when R is rotated about the y-axis?

  • 25.447

  • 98.960

  • 122.400

  • 384.531

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