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

Volumes of Revolution (College Board AP® Calculus AB)

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Volumes of Revolution

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Volumes of Revolution

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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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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4
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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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5
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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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