The Big Idea
When a solid has a known cross-sectional shape, you can find its volume by imagining the solid as a collection of extremely thin slices. Each slice has a certain area, and the integral adds those areas across the interval.
The key formula is:
The challenging part is usually not the integration itself. The real AP Calculus skill is determining the cross-sectional area correctly from the geometry described in the problem.
The AP Calculus Method
- Identify the base region. Determine the curves or boundaries that enclose the region.
- Identify the direction of the cross-sections. Decide whether the slices are perpendicular to the -axis or the -axis.
- Find the length of the segment in the base. This becomes the relevant side, diameter, or radius of the cross-section.
- Use the stated cross-sectional shape to write its area as a function of the variable.
- Integrate the cross-sectional area. This gives the volume.
Cross-Section Formulas You Should Know
For a square whose side length is :
For a rectangle with side lengths and :
For a semicircle with diameter , the radius is , so:
These formulas are then combined with the length of the base segment to create the function .
Worked Examples
The base of a solid is the region bounded by and . Cross-sections perpendicular to the -axis are squares. Find the volume.
The base is the region bounded by and . Cross-sections perpendicular to the -axis are semicircles. Find the volume.
The base is the region under and above the -axis for . Cross-sections perpendicular to the -axis are rectangles whose height is twice their width. Find the volume.
The region is bounded by , the -axis, and . Cross-sections perpendicular to the -axis are squares. Find the volume.
How to Recognize the Correct Setup
On AP Calculus questions, the geometry description tells you exactly what your cross-sectional area should look like. A useful habit is to stop before integrating and write a sentence identifying the segment that becomes the defining measurement of the cross-section.
For example, if the problem says that cross-sections are squares perpendicular to the -axis, use the vertical distance between the curves. If the problem says the cross-sections are perpendicular to the -axis, use the horizontal distance instead.
Common Mistakes
For vertical slices, the length of the base segment is usually . For horizontal slices, it is usually . Mixing these up gives the wrong cross-sectional area before you even start integrating.
If the given segment is the diameter , do not put directly into the circle-area formula. The radius is .
If the side length is , the cross-sectional area is , not .
The integral must contain the area of a cross-section. Do not integrate the distance between the curves by itself unless that distance actually represents the cross-sectional area.
If the cross-sections are perpendicular to the -axis, integrate with respect to . If they are perpendicular to the -axis, integrate with respect to .
Practice Problems
The base of a solid is bounded by and . Cross-sections perpendicular to the -axis are squares. Find the volume.
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The base is the region under and above the -axis on . Cross-sections perpendicular to the -axis are semicircles. Find the volume.
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The base is bounded by , , and . Cross-sections perpendicular to the -axis are squares. Find the volume.
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The base is under and above the -axis for . Cross-sections perpendicular to the -axis are rectangles whose height is three times their width. Find the volume.
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The base is bounded by , , and . Cross-sections perpendicular to the -axis are squares. Find the volume.
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The base is bounded by and . Cross-sections perpendicular to the -axis are semicircles with diameters lying in the base. Find the volume.
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The base is bounded by and . Cross-sections perpendicular to the -axis are equilateral triangles. Find the volume.
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A solid has a base given by the region between and . Cross-sections perpendicular to the -axis are rectangles. The height of each rectangle is half its base. Write, but do not evaluate, an integral representing the volume.
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A Reliable AP Exam Checklist
- What direction are the slices? Perpendicular to or ?
- What is the length of the segment in the base?
- What shape is the cross-section?
- What is the area formula for that shape?
- Have you substituted the correct segment length into the area formula?
- What are the correct limits?
- Does the final integral represent area accumulated across the entire base?
The most important conceptual point is that volume is accumulated cross-sectional area. Once you can correctly translate the geometry into , the calculus is usually straightforward.
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