Cross-sectional solids appear in almost every AP® Calculus free-response set. These solids are built by stacking thin slices, and the College Board labels the skill as “8.7 Volumes with Cross Sections.” Therefore, mastering the cross sectional area formula is essential.

After reading, students should be able to:

• link familiar 2-D area formulas to definite integrals
• set up and evaluate volumes when the slices are squares or rectangles

The Big Idea: From Slices to Volume

What is a cross section?

Picture cutting a loaf of bread. Each slice is a 2-D shape called a cross section. The loaf’s volume comes from adding the volumes of all slices.

Turning a 2-D area into a 3-D volume

A single slice is extremely thin. Its volume equals “area of the face” times “thickness dx.” Summing (integrating) those pieces produces the full volume.

The general cross sectional area formula

V=\int_{a}^{b} A(x)dx  or V=\int_{c}^{d} A(y)dy

Here, A(x) is the area of one slice perpendicular to the x-axis, and [a,b] are the x-bounds of the base region.

Graphing insight

On a graph, draw the base region in the xy-plane. Next, sketch a thin vertical “slice.” That slice lifts straight up out of the page, creating the 3-D shape.

Square Cross Sections over a Region

Setting up the slice

When cross sections are squares, each slice’s face is a square.

Area of a square

A=s^{2}

Linking the side length s to the graph

If slices are perpendicular to the x-axis, the side length equals top curve − bottom curve.

Example 1: Squares between two parabolas

1. Base region: between y=x^{2} and y=2x on 0\le x\le2.
2. Side length: s=2x-x^{2}.
3. Area of one slice:
   • A(x)=s^{2}=(2x-x^{2})^{2}=4x^{2}-4x^{3}+x^{4}
4. Volume integral:
   • V=\int_{0}^{2}\bigl(4x^{2}-4x^{3}+x^{4}\bigr)dx
5. Evaluate:
   • \begin{aligned}V&=\left[\frac{4}{3}x^{3}-x^{4}+\frac{1}{5}x^{5}\right]_{0}^{2}\\[4pt]&=\frac{32}{3}-16+\frac{32}{5}\\[4pt]&=\frac{16}{15}\text{ unit}^{3}\end{aligned}
6. Graph insight: The slice rests between the two curves, so each square rises straight up. The width shrinks where the curves approach each other.

Rectangular Cross Sections (Rectangle Cross Section Formula)

Basic area

A=\text{base}\times\text{height}

Common exam cue

Often the prompt states, “height is k times the base.” That single sentence is the key to writing A(x).

Integrating rectangles

Once an expression for base is found, multiply by k for the height, substitute into the cross sectional area formula, and integrate.

Example 2: Rectangles under a square-root curve

1. Base region: under y=\sqrt{x} from x=0 to x=4.
2. Base of each rectangle: b=\sqrt{x}.
3. Height is three times the base: h=3\sqrt{x}.
4. Area of a slice:
   • A(x)=b\cdot h=\sqrt{x}(3\sqrt{x})=3x
5. Volume integral:
   • V=\int_{0}^{4} 3xdx
6. Evaluate:
   • V=3\left[\frac{x^{2}}{2}\right]_{0}^{4}=3\left(\frac{16}{2}\right)=24\text{ unit}^{3}
7. Graph insight: Each rectangle stands upright; because height = 3 × base, the top edge rises steeply, giving the solid a “fans out” appearance.

Mixed Shapes & Strategy Tips

• Identify the slicing axis first.
• Express every slice dimension in one variable (x or y), never both.
• Write the correct area formula (square, rectangle, semicircle, etc.).
• Set correct bounds. Check the intersection points carefully.
• Evaluate the integral analytically when possible; otherwise, use a calculator.

Common AP® pitfalls

• Forgetting units in the final answer
• Mixing up top and bottom for s
• Ignoring constants like “height is k times base”
• Accidentally slicing parallel to the wrong axis

Quick Reference Chart – Key Vocabulary

Term	Definition	Sample Formula
Cross section	A 2-D slice taken perpendicular to an axis	—
Cross sectional area formula	Integral that adds all slice areas to find volume	V=\int A(x)dx
Square slice	Slice whose face is a square	A=s^{2}
Rectangle cross section formula	Slice face is rectangle; use base × height	A=b\cdot h

Conclusion

The cross sectional area formula transforms basic geometry into a powerful calculus tool. By writing A(x) for each slice and integrating, any square or rectangular solid can be handled with ease. Therefore, consistent practice with different slice shapes—especially squares and the rectangle cross section formula—builds the confidence needed for College Board skill 8.7 volumes with cross sections. Keep slicing, integrating, and checking those bounds, and solid-volume questions will soon feel routine.

Sharpen Your Skills for AP® Calculus AB-BC

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• 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
• 8.8 Volumes with Cross Sections: Triangles and Semicircles

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