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

Volumes with Cross Sections (College Board AP® Calculus AB)

Exam Questions

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Volumes with Cross Sections

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Volumes with Cross Sections

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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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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3
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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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4
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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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5
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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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