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Find the Volume for Solids with Known Cross Sections for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

If A(x) is the area of a cross section of a solid and A(x) is continuous on [a, b], then the volume of the solid from x = a to x = b is:

See Figure 12.4-1.

Solids with Known Cross Sections

Note: A cross section of a solid is perpendicular to the height of the solid.

Example 1

The base of a solid is the region enclosed by the ellipse . The cross sections are perpendicular to the x-axis and are isosceles right triangles whose hypotenuses are on the ellipse. Find the volume of the solid. See Figure 12.4-2.

Solids with Known Cross Sections

Step 1.   Find the area of a cross section A(x ).

Step 2.   Set up an integral.

Step 3.   Evaluate the integral.

    The volume of the solid is
    Verify your result with a graphing calculator.

Example 2

Find the volume of a pyramid whose base is a square with a side of 6 feet long, and a height of 10 feet. See Figure 12.4-3.

Step 1.   Find the area of a cross section A(x ). Note each cross section is a square of side 2s.

    Similar triangles:

Step 2.   Set up an integral.

Step 3. Evaluate the integral.

The volume of the pyramid is 120 ft3.

Example 3

The base of a solid is the region enclosed by a triangle whose vertices are (0, 0), (4, 0) and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Using a calculator, find the volume of the solid. (See Figure 12.4-4.)

Solids with Known Cross Sections

Step 1.   Find the area of a cross section.

    Equation of the line passing through (0, 2) and (4, 0):

Step 2.   Set up an integral.

Step 3.   Evaluate the integral.

Thus the volume of the solid is 2.094.

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

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