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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Graph showing the shaded region R bounded by the y-axis, the line x=3, and the curves y=e^cosx and y=sqrt(x)-3

Let R be the region enclosed by the graphs of f open parentheses x close parentheses equals e to the power of cos x end exponent and g open parentheses x close parentheses equals square root of x minus 3, the y-axis, and the vertical line x equals 3, as shown in the figure above.

Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y equals 3.

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2
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Graph showing the shaded region R in the first quadrant bounded by the positive coordinate axes and the curve y=((x-2)/2)^2

Let R be the region in the first quadrant enclosed by the coordinate axes and the graph of y equals open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses squared, as shown in the figure above.

Find the volume of the solid generated when R is rotated about the x-axis.

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3
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A graph of the curve with equation x^2=(ky)^2*(9-y^2) for x>0, y>0

A company designs fishing lures using the family of curves with equation open parentheses k y close parentheses squared open parentheses 9 minus y squared close parentheses minus x squared equals 0, where k is a positive constant. The figure above shows the region in the first quadrant bounded by the y-axis and the curve open parentheses k y close parentheses squared open parentheses 9 minus y squared close parentheses minus x squared equals 0, for some k. Each fishing lure is in the shape of the solid generated when such a region is revolved about the y-axis. Both x and y are measured in inches.

For a particular fishing lure, the volume is fraction numerator 162 pi over denominator 25 end fraction cubic inches. What is the value of k for this fishing lure?

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4
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Graph of the shaded region R bounded by the y-axis and the curves y=sec(1-x) and y=x^3/8

Let R be the region in the first quadrant enclosed by the y-axis, the line x equals 2, and the graphs of y equals sec open parentheses 1 minus x close parentheses and y equals x cubed over 8, as shown in the figure above.

Find the volume of the solid generated when R is rotated about the x-axis.

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5
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Graph of the shaded region R bounded by the coordinate axes, the line y=8, and the curves with equations y=x^2+1 and y=(x-4)^3

Let R be the region in the first quadrant enclosed by the coordinate axes, the line y equals 8, and the graphs of y equals x squared plus 1 and y equals open parentheses x minus 4 close parentheses cubed, as shown in the figure above.

Find the volume of the solid generated when R is rotated about the y-axis.

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