Volumes with Cross-Sections: Triangles and Semicircles

What Cross-Section Volume Problems Are Really Asking

A volume problem with cross-sections gives you a two-dimensional region in the -plane and tells you what shape appears when the solid is sliced perpendicular to an axis. Your job is to determine the area of one representative cross-section and then add up those infinitely many thin slices with a definite integral.

The general idea is simple:

The difficult part is usually not the integration. It is finding , the area of a typical cross-section, in terms of the variable of integration.

Once you know the cross-sectional area, the volume is just the accumulation of those areas across the interval.

The Basic Setup

Suppose the base region extends from to , and a cross-section perpendicular to the -axis has area . Then the volume is:

For cross-sections perpendicular to the -axis, the setup becomes:

So there are really two separate tasks:

1. Find the length of the segment that becomes a side, base, or diameter of the cross-section.
2. Use the geometry formula for the specified cross-section to find its area.

Cross-Sections That Are Equilateral Triangles

If the cross-section is an equilateral triangle, every side has the same length. If that side length is , the area of the triangle is:

In a volume problem, the important question is: what does the base region tell us about ?

If the vertical distance between two curves is the side length of each triangular cross-section, then:

Notice that you do not integrate the side length directly. You first convert the side length into the area of the cross-section.

Cross-Sections That Are Semicircles

For a semicircular cross-section, the segment from the lower curve to the upper curve is often the diameter of the semicircle.

If the diameter is , then the radius is . The area of the semicircle is therefore:

Substituting the diameter gives a particularly useful formula:

This is an extremely common AP Calculus setup. If the distance between two curves is , then that distance is the diameter, not the radius. The factor of is where many errors occur.

A Reliable Four-Step Method

1. Identify the direction of the cross-sections. Decide whether the slices are perpendicular to the -axis or -axis.
2. Find the segment length. Determine the distance between the two boundary curves at a typical value of the integration variable.
3. Convert the segment into cross-sectional area. Use the appropriate geometry formula for the triangle, semicircle, square, rectangle, or other specified shape.
4. Integrate the cross-sectional area. Add all the cross-sectional areas over the required interval.

Worked Examples

Triangle vs. Semicircle: The Key Difference

The same base region can produce very different volumes depending on the shape of the cross-section. The segment length coming from the graph may be exactly the same, but the formula for cross-sectional area changes.

So if the distance between the curves is , an equilateral triangle gives an area proportional to , while a semicircle gives an area proportional to .

How to Recognize the Diameter or Side Length

This is one of the most important reading skills for these problems. Look at the wording carefully.

• If the problem says the cross-sections are equilateral triangles and the segment in the base becomes the side of the triangle, that segment is the triangle's side length.
• If the problem says the cross-sections are semicircles and the segment in the base becomes the diameter, use the segment as the diameter.
• Do not automatically call every segment a radius. For semicircles, determine whether the problem gives you the diameter or the radius before calculating area.

Common Mistakes

A Fast AP Exam Strategy

On an AP free-response question, you can make these problems much easier by writing down the cross-sectional area before worrying about the integral.

1. Draw or visualize one slice.
2. Label the segment from the base region.
3. Decide whether that segment is a side, base, diameter, or radius.
4. Write the cross-sectional area formula.
5. Substitute the graph-derived length.
6. Integrate over the correct interval.

If you get stuck, don't immediately try to integrate. The calculus is usually the final step. The geometry setup is the real heart of the problem.

Practice Problems