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

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

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Volumes of Revolution

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1
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1 mark

A solid metal sculpture has the shape obtained by revolving the curve y=3+cos 2x from x=0 to x=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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1 mark
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=4, the line x=π3, and the curve y=4 sin x, as shown in the figure above. A solid is generated by rotating R about the line y=4.

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

  • π∫0π3(4−4 sin x) dx

  • π∫0π3(4−4 sin x)2 dx

  • π∫04(4−4 sin x) dx

  • π∫04(4−4 sin x)2 dx

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3
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1 mark
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=3, to the right by the y=ex, 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?

  • π∫13e2y dy

  • π∫13ln2y dy

  • π∫0ln 3e2y dy

  • π∫0ln 3ln2y dy

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4
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1 mark
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=x2+2, the y-axis and the lines y=1 and x=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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1 mark
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=x2, the lines y=9 and x=−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?

  • π∫09y−1 dy

  • π∫19y−1 dy

  • π∫−3081−x4 dx

  • π∫−3−181−x4 dx

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1
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1 mark

A region in the first quadrant is enclosed by the graphs of y=x3 and y=x. If the region is rotated about the x-axis, which of the following represents the volume of the solid that is generated?

  • ∫01(x−x3) dx

  • ∫01π(x−x3)2 dx

  • ∫01πx(1−x5) dx

  • ∫01πx72 dx

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2
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1 mark

Let R be the region in the first quadrant enclosed by the coordinate axes, the line y=4, and the curve with equation y=x2−3, 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?

  • 23(77−33)π

  • 28π3

  • 20π

  • 1003π y2+3

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3
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1 mark

What is the volume of the solid generated when the region bounded by the graph of x=y−1 and the lines x=0 and y=7 is revolved about the y-axis?

  • 18.000

  • 30.781

  • 54.978

  • 56.549

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4
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1 mark

Let R be the region bounded by the graphs of y=2x, y=24, and x=0. Which of the following finds the volume of the solid formed by revolving R about the line y=−3?

  • π∫04 (24+3)2−(2x+3)2 dx

  • π∫04 (24−2x−3)2 dx

  • π∫04 (24−2x+3)2 dx

  • π∫04 (24−2x)2+3 dx

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5
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1 mark
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=x5, the line y=6−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=6.

  • 14π

  • 22π

  • 430π

  • 864π

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1
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1 mark
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 x2+4y2=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

  • 32π3

  • 64π3

  • 32π

  • 48π

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2
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1 mark

Let R be the region enclosed by the graph of f(x)=arcsinx, the line y=π2, 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=1?

  • 1.119

  • 1.215

  • 2.467

  • 3.816

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3
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1 mark

A region in the first quadrant is enclosed by the graphs of y=e3x, x=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?

  • 2π∫01xe3x dx

  • π∫01e6x dx

  • π9∫1e3ln2y dy

  • πe3−π9∫1e3ln2y dy

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4
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1 mark
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=sin x as shown in the diagram above.

A solid is generated by revolving R about the line y=−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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1 mark
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=x, and the line y=17(x+10) 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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