Topic 8.8 Notes – Volumes with Cross Sections: Triangles and Semicircles

How volumes with known cross sections work

The structure is always the same:

$$V = \int_a^b A(x)\,dx$$

• $A(x)$ is the area of one cross section
• $[a,b]$ comes from where the base region exists
• The slices are perpendicular to the axis named in the problem

You already know this idea from Riemann sums. The only difference is that instead of thin rectangles, you’re stacking actual geometric shapes.

The key connection:

• The base is a region between curves.
• The length of each slice comes from the distance between those curves.
• That length becomes a side, leg, or diameter of your cross section.

Everything flows from finding that length correctly.

The cross-section shapes you must know

These are the standard AP shapes. Know the area formulas and how the distance between curves fits in.

Equilateral triangles

Area formula:

$$A = \frac{\sqrt{3}}{4}s^2$$

• $s$ is the side length.
• In these problems, $s = \text{top} - \text{bottom}$ (if vertical slices).

So:

$$A(x) = \frac{\sqrt{3}}{4}[\text{top} - \text{bottom}]^2$$

Right isosceles triangles

Area formula:

$$A = \frac{1}{2}s^2$$

• $s$ is the length of the equal legs.
• Usually that equals the distance between curves.

So:

$$A(x) = \frac{1}{2}[\text{top} - \text{bottom}]^2$$

Semicircles

Full circle area is $\pi r^2$.
Semicircle area is:

$$A = \frac{1}{2}\pi r^2$$

Important detail:

• The distance between curves is usually the diameter, not the radius.
• So $r = \frac{1}{2}(\text{top} - \text{bottom})$.

That gives:

$$A(x) = \frac{\pi}{8}[\text{top} - \text{bottom}]^2$$

Notice the pattern. Almost always you end up with:

constant × (distance between curves)²

Setting up the volume step by step

When this shows up on a quiz or FRQ, move in this order.

1. Find the bounds

Set the curves equal to find intersection points.
Those x-values are your limits of integration.

If you’re unsure which function is bigger, plug in a test value.

2. Find the slice length

This is the most important step.

• If slices are perpendicular to the x-axis, use vertical distance
  → $\text{top} - \text{bottom}$
• If perpendicular to the y-axis, use horizontal distance
  → $\text{right} - \text{left}$

Your variable must match the direction of slicing.

3. Build the area function

Plug that length into the correct geometric formula.

For example, if cross sections are semicircles:

1. Diameter = top − bottom
2. Radius = half of that
3. Plug into $\frac{1}{2}\pi r^2$

Then integrate:

$$V = \int_a^b A(x)\,dx$$

Sometimes the AP only asks for the integral setup, not the evaluated number. Don’t overdo algebra if they don’t ask for it.

How this appears on the AP exam

Common wording:

• “The region bounded by $f$ and $g$ forms the base…”
• “Cross sections perpendicular to the x-axis are equilateral triangles.”
• “Find the volume.”

Occasionally:

• They give you $A(x)$ directly and just want $\int A(x)\,dx$.
• A calculator-active FRQ may expect numerical evaluation.
• No-calculator questions usually keep the algebra manageable.

One subtle grading point: if you forget to divide by 2 for a semicircle radius, you lose more than just arithmetic credit because your setup is wrong.

Common mistakes that cost points

• Forgetting diameter vs radius in semicircles. This changes the answer by a factor of 4.
• Squaring only part of the expression. It must be $(\text{top} - \text{bottom})^2$.
• Mixing up which function is on top.
• Using $dx$ when the slices are horizontal and should be $dy$.

Most errors here are algebra or setup errors, not calculus errors.