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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 8: Applications of IntegrationVolumes with Cross Sections

Volumes with Cross Sections (College Board AP® Calculus BC): Exam Questions

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Volumes with Cross Sections

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1
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The area, in square meters, of the the vertical cross section of a fuel tank at a distance of x meters from one end of the tank is modeled by the function f given by f open parentheses x close parentheses equals fraction numerator square root of x over denominator e to the power of x end fraction. The tank has a length of 3 meters.

Based on this model, what is the volume of the tank in cubic meters?

  • 0.787

  • 0.913

  • 1.470

  • 3.142

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2
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The base of a solid is the region in the first quadrant bounded by the y-axis, the x-axis, the graph of y equals 2 e to the power of x, and the vertical line x equals 0.5. For this solid, each cross-section perpendicular to the x-axis is a square. What is the volume of the solid?

  • e over 2 minus 1 half

  • 2 e

  • 8 e minus 8

  • 2 e minus 2

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3
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The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the curve y equals sin space x, and the vertical line x equals pi over 3 . If cross sections of the solid perpendicular to the x-axis are semicircles, which integral describes the volume of the solid?

  • integral subscript 0 superscript pi over 3 end superscript space pi over 2 space sin squared italic space x italic space italic d x

  • integral subscript 0 superscript pi over 3 end superscript space pi over 8 space sin squared italic space x italic space italic d x

  • integral subscript 0 superscript pi over 3 end superscript space pi over 4 space sin italic space x italic space italic d x

  • integral subscript 0 superscript pi over 3 end superscript space pi over 8 space sin italic space x italic space italic d x

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4
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The shaded region R, shown below is bounded by the x-axis, the y-axis, the function f open parentheses x close parentheses and the line x equals 1.

Graph showing a parabola intersecting a vertical line, with the area between them shaded

The region forms the base of a solid. If cross sections of the solid perpendicular to the x-axis are equilateral triangles, which integral describes the volume of the solid?

  • 1 half integral subscript 0 superscript 1 open parentheses f open parentheses x close parentheses close parentheses squared space italic d x

  • fraction numerator square root of 3 over denominator 2 end fraction integral subscript 0 superscript 1 open parentheses f open parentheses x close parentheses close parentheses squared space italic d x

  • fraction numerator square root of 3 over denominator 4 end fraction integral subscript 0 superscript 1 open parentheses f open parentheses x close parentheses close parentheses squared space italic d x

  • fraction numerator square root of 3 over denominator 8 end fraction integral subscript 0 superscript 1 open parentheses f open parentheses x close parentheses close parentheses squared space italic d x

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1
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Let R be the region bounded below by the graph of y equals cos x and above by the graph of y equals sin x, between x equals pi over 4 and x equals fraction numerator 5 pi over denominator 4 end fraction. R is the base of a solid whose cross sections perpendicular to the x-axis are squares. What is the volume of the solid?

  • 1.414

  • 2.828

  • 3.142

  • 5.312

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2
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Graph showing a shaded region bounded by the line 3x+5y=15 and the positive x- and y-axes

The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line 3 x plus 5 y equals 15, as shown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

  • 3.750

  • 5.890

  • 11.781

  • 23.562

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3
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The base of a solid is the region in the first and fourth quadrants bounded by the lines y equals x and y equals negative x between x equals 0 and x equals 5. If cross sections of the solid perpendicular to the x-axis are rectangles, with the height of each rectangle equal to one half of its base, which of the following integrals would correctly calculate the volume of the solid?

  • integral subscript 0 superscript 5 2 x space d x

  • integral subscript 0 superscript 5 2 x squared space d x

  • integral subscript 0 superscript 5 pi x squared space d x

  • integral subscript 0 superscript 5 4 x squared space d x

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4
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The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y equals arctan open parentheses 2 x close parentheses, the horizontal line y equals 4, and the vertical line x equals 1.5. For this solid, each cross-section perpendicular to the x-axis is a square. What is the volume of the solid?

  • 4.702

  • 14.919

  • 22.698

  • 32.118

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5
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Let R be the shaded region in the first quadrant bounded by the graphs of y equals 2 cos open parentheses fraction numerator pi x over denominator 2 end fraction close parentheses and y equals open parentheses x minus 3 over 2 close parentheses squared minus 1 fourth, as shown in the figure below. The region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in region R. What is the volume of the solid?

Graph showing two intersecting curves, shaded region between them, and points of intersection marked at (0,2) and (1,0)

  • 0.117

  • 0.220

  • 0.234

  • 0.440

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1
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The base of a solid is the region in the first quadrant bounded by the curves y equals x squared and y equals square root of x. If cross sections of the solid perpendicular to the x-axis are triangles, with the height of each triangle equal to its x-coordinate, which of the following integrals would correctly calculate the volume of the solid?

  • integral subscript 0 superscript 1 open parentheses x squared minus square root of x close parentheses squared space d x

  • integral subscript 0 superscript 1 open parentheses square root of x minus x squared close parentheses space d x

  • integral subscript 0 superscript 1 x over 2 open parentheses x squared minus square root of x close parentheses space d x

  • integral subscript 0 superscript 1 x over 2 open parentheses square root of x minus x squared close parentheses space d x

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2
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The base of a solid is the region enclosed by the circle given by x squared plus y squared equals r squared. If cross sections of the solid perpendicular to the x-axis are squares, calculate the volume of the solid in terms of r.

  • 4 over 3 r cubed

  • 8 over 3 r cubed

  • 16 over 3 r cubed

  • 8 r cubed

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3
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Let R be the shaded region in the first quadrant bounded by the graphs of y equals f open parentheses x close parentheses, y equals g open parentheses x close parentheses, and the line x equals 1, as shown in the diagram below.

The functions f and g are defined as f open parentheses x close parentheses equals 4 e to the power of x and g open parentheses x close parentheses equals 4 e to the power of negative x end exponent.

The region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is an isosceles right triangle with the hypotenuse in region R. Which integral describes the volume of the solid?

Graph showing two intersecting curves with a shaded region between them, bounded by a vertical line and the y-axis

  • 8 integral subscript 0 superscript 1 open parentheses e to the power of 2 x end exponent minus e to the power of negative 2 x end exponent minus 2 close parentheses space d x

  • 8 integral subscript 0 superscript 1 open parentheses e to the power of 2 x end exponent plus e to the power of negative 2 x end exponent minus 2 close parentheses space d x

  • 4 integral subscript 0 superscript 1 open parentheses e to the power of 2 x end exponent minus e to the power of negative 2 x end exponent minus 2 close parentheses space d x

  • 4 integral subscript 0 superscript 1 open parentheses e to the power of 2 x end exponent plus e to the power of negative 2 x end exponent minus 2 close parentheses space d x

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