Volumes with Cross-Sections: Squares and Rectangles

What Cross-Section Problems Are Really Asking

In a standard volume problem, you may already know the shape of the solid: disks, washers, or cylindrical shells. Cross-section problems are different. Instead of giving you the three-dimensional solid directly, the problem tells you what shape each slice of the solid has.

For example, a problem might say that a region in the -plane is the base of a solid whose cross-sections perpendicular to the -axis are squares. Your job is to determine the side length of each square from the base region, find its area, and then integrate those areas.

The entire strategy can be summarized in one sentence:

The Basic Idea: Slice the Solid

Imagine cutting the solid into extremely thin slices. If the slices are perpendicular to the -axis, then each slice has a small thickness and a cross-sectional area .

The volume of one very thin slice is approximately:

Adding infinitely many of these thin slices gives the exact volume:

So the calculus itself is usually not the difficult part. The real work is determining what the cross-sectional area should be.

A Step-by-Step Method

1. Identify the direction of the cross-sections. Perpendicular to the -axis means the slices are vertical. Perpendicular to the -axis means the slices are horizontal.
2. Find the length of the segment in the base region. This segment becomes the side length, width, or another dimension of the cross-section.
3. Write the cross-sectional area. For a square, use side squared. For a rectangle, multiply length by width.
4. Set up the volume integral. Integrate the cross-sectional area over the correct interval.
5. Evaluate the integral. The result is the volume of the solid.

Squares: The Most Important Case

Suppose each cross-section is a square. If the side length of the square is , then its area is:

This means that once you determine the length of the corresponding segment in the base region, the cross-sectional area follows immediately.

If the cross-sections are perpendicular to the -axis and the top and bottom curves are and , then the vertical distance between the curves is:

That distance is the side length of the square. Therefore:

And the volume is:

Worked Examples

How to Handle Rectangles

Rectangular cross-sections require one additional piece of information: knowing one side of the rectangle is not enough. You must determine both dimensions.

If the segment in the base has length and the problem says the rectangle has height , then:

For example, if the height is three times the base, then:

Substitute the appropriate expression for based on the geometry described in the problem.

The Direction of the Slice Matters

One of the most common AP Calculus mistakes is integrating with respect to the wrong variable.

So:

• Cross-sections perpendicular to the x-axis → vertical slices → integrate with respect to .
• Cross-sections perpendicular to the y-axis → horizontal slices → integrate with respect to .

When the slices are horizontal, you will often need to rewrite the curves as in terms of . This is why some cross-section problems look more complicated than ordinary volume problems.

Common Mistakes

A Fast AP Exam Checklist

1. What shape are the cross-sections? Square? Rectangle? Triangle? Semicircle?
2. Which direction are the slices? Perpendicular to the -axis or -axis?
3. What segment in the base determines the dimensions?
4. What is the area of one cross-section?
5. What variable should you integrate with respect to?
6. What are the limits of integration?
7. Does the final answer have cubic units?

Practice Problems

The Big Picture

Cross-section volume problems are really an application of one fundamental idea: volume is accumulated cross-sectional area.

The geometry tells you what is. The calculus tells you how to accumulate those areas. Once you can move confidently from the base region to the cross-sectional area, these problems become much more systematic.

For AP Calculus, remember the sequence: