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Volumes with
Cross Sections

Learn to compute volumes of solids whose cross sections are squares and rectangles, using definite integrals.

AP Calculus AB

What Is a Solid with Known Cross Sections?

Imagine a loaf of bread: each slice has a recognizable shape. A solid with known cross sections is a 3D object where every slice perpendicular to a chosen axis has a shape you can describe: a square, rectangle, semicircle, or triangle.

Visualizing the 3D Solid

Base region on the x-axis slice slice slice a b dx

Each green slice is a cross section. Its area depends on where you cut along the \(x\)-axis. We sum up infinitely many thin slices of width \(dx\) to find the total volume.

Why Cross Sections Matter

Beyond Revolution
Not every solid is formed by revolving a curve. Cross sections let you handle solids with flat, polygonal slices.
Same Core Idea
Volume = accumulation of area. Whether a disk, washer, or square slice: \(V = \int A(x)\,dx\).
AP Exam Staple
Cross-section problems appear almost every year on the AP Calculus AB free-response section.

The Volume Formula

The key challenge is expressing \(A(x)\) in terms of \(x\). For square and rectangle cross sections, the area formula depends on the side length, which is determined by the base region.

Square Cross Sections

x-axis (base line) \(s(x)\) square slice x a b \(f(x)\)

The side length \(s(x)\) is typically the vertical distance between two curves: \(s(x) = f(x) - g(x)\).

Rectangle Cross Sections

\(w(x)\) rectangle slice \(h(x)\) x

Often the problem states height as a multiple of width, e.g. "height is twice the base": \(h(x) = 2\,w(x)\).

Squares vs. Rectangles

Square Cross Section Rectangle Cross Section
Shape All four sides equal \(w \neq h\) (usually)
Side from base \(s(x) = f(x) - g(x)\) \(w(x) = f(x) - g(x)\)
Area formula \(A(x) = [s(x)]^2\) \(A(x) = w(x) \cdot h(x)\)
Extra info needed? No: side = base width Yes: height relation given

How to Solve These Problems

1
Sketch the base region. Identify the bounding curves and their intersection points to find \([a, b]\).
2
Determine the side/width. The distance across the base at position \(x\) is \(s(x) = f(x) - g(x)\).
3
Write \(A(x)\). For squares: \([s(x)]^2\). For rectangles: \(w(x) \cdot h(x)\), substituting the given height relationship.
4
Integrate. Compute \(\displaystyle V = \int_a^b A(x)\,dx\) using standard integration techniques.
5
Check units and sign. Volume must be positive. If the answer is negative, re-check which curve is on top.

Example 1: Square Slices

The base of a solid is the region bounded by \(y = \sqrt{x}\) and \(y = 0\) from \(x = 0\) to \(x = 4\). Cross sections perpendicular to the \(x\)-axis are squares. Find the volume.

\(s(x)\) 4 0 x y \(y = \sqrt{x}\)

Example 1: Solution

1
Side length: \(s(x) = \sqrt{x} - 0 = \sqrt{x}\)
2
Cross-sectional area: \(A(x) = [\sqrt{x}]^2 = x\)
3
Integrate: \(\displaystyle V = \int_0^4 x\,dx = \left[\frac{x^2}{2}\right]_0^4 = \frac{16}{2} = 8\)

Example 2: Rectangle Slices

The base is bounded by \(y = x^2\) and \(y = 1\). Cross sections perpendicular to the \(y\)-axis are rectangles whose height is 3 times the base width. Find the volume.

\(y = 1\) \(w(y) = 2\sqrt{y}\) x y \(y = x^2\)

Example 2: Solution

1
Rewrite for \(y\): \(y = x^2 \Rightarrow x = \pm\sqrt{y}\). The width is \(w(y) = 2\sqrt{y}\). Integration bounds: \(y \in [0, 1]\).
2
Height: \(h(y) = 3 \cdot w(y) = 6\sqrt{y}\).
3
Area: \(A(y) = 2\sqrt{y} \cdot 6\sqrt{y} = 12y\).
4
Integrate: \(\displaystyle V = \int_0^1 12y\,dy = \left[6y^2\right]_0^1 = 6\).

Example 3: Two-Curve Base

The base is bounded by \(y = x\) and \(y = x^2\). Cross sections perpendicular to the \(x\)-axis are squares. Find the volume.

\(s(x)\) \(y = x\) \(y = x^2\) 0 1

Common Mistakes

✕ Forgetting to Square
For square slices, you need \([s(x)]^2\), not just \(s(x)\). Integrating the side length alone gives area of the base, not volume.
✕ Wrong Axis
If slices are perpendicular to the \(y\)-axis, express everything in terms of \(y\) and use \(dy\). Don't mix axes.
✕ Top Minus Bottom
Always subtract: \(f(x) - g(x)\) where \(f \ge g\). Reversing the order gives a negative side length, which squares away but masks errors in rectangle problems.

Summary Cheat Sheet

Step Square Rectangle
Side / Width \(s = f(x) - g(x)\) \(w = f(x) - g(x)\)
Height \(h = s\) (same as side) \(h\) given by problem
Area \(A(x)\) \(s^2\) \(w \cdot h\)
Volume \(\displaystyle\int_a^b s^2\,dx\) \(\displaystyle\int_a^b w \cdot h\,dx\)
Key check Did you square the side? Did you substitute for \(h\)?

Worked practice: Section 9, Practice Problem 4 — Applied

Work through this original lesson problem, then reveal the solution and illustrations.

A landscape architect designs a decorative garden bed whose footprint in the xy-plane is the region R bounded by y = 6 − x² and y = 2. The bed is filled with soil so that cross sections perpendicular to the y-axis are squares. (a) Sketch the region R and identify the limits of integration in terms of y. (1 point) (b) Express the side length of a square cross section as a function of y. (1 point) (c) Write and evaluate the integral that gives the volume of soil needed. (2 points)

Show answer and explanation

Rubric (4 points): Point 1 (Part a): The region R is bounded above by the parabola y = 6 − x² and below by the horizontal line y = 2. Setting 6 − x² = 2 gives x = ±2. When slicing perpendicular to the y-axis, y ranges from 2 to 6. Award 1 point for correct limits y = 2 to y = 6 and correct identification of the bounding curves. Point 2 (Part b): From y = 6 − x², solve for x: x = ±√(6 − y). At height y, the horizontal extent of the region is from x = −√(6 − y) to x = √(6 − y), so the side length of the square is s(y) = 2√(6 − y). Award 1 point. Point 3 (Part c — integral setup): A(y) = [2√(6 − y)]² = 4(6 − y). So V = ∫₂⁶ 4(6 − y) dy. Award 1 point for correct area function and integral. Point 4 (Part c — evaluation): V = 4∫₂⁶ (6 − y) dy = 4[6y − y²/2]₂⁶ = 4[(36 − 18) − (12 − 2)] = 4[18 − 10] = 4(8) = 32 cubic units. Award 1 point for correct numerical answer.

Square cross sections above a parabolic base. The section at y=4 has side 2√2 and area 8; the cutaway shows y=4 to 6 while the full y=2 to 6 solid has volume 32.
Square sections perpendicular to the y-axis have area 4(6−y), producing a 32-cubic-unit solid.
Base bounded by y=6−x² and y=2, with x endpoints −2 and 2. The section at y=4 spans width 2√2.
Base bounded by y=6−x² and y=2, with x endpoints −2 and 2. The section at y=4 spans width 2√2.

Question 1 of 5

The base of a solid is the region bounded by \(y = 4 - x^2\) and \(y = 0\). Cross sections perpendicular to the \(x\)-axis are squares. Which integral gives the volume?

Question 2 of 5

A solid has base bounded by \(y = \sqrt{x}\) and \(y = 0\) for \(0 \le x \le 9\). Cross sections perpendicular to the \(x\)-axis are rectangles with height twice the base. What is the volume?

Question 3 of 5

For the region bounded by \(y = x\) and \(y = x^2\) with square cross sections perpendicular to the \(x\)-axis, \(A(x) =\)

Question 4 of 5

Cross sections perpendicular to the \(y\)-axis are squares. The base is bounded by \(x = y^2\) and \(x = 4\). What are the correct limits of integration?

Question 5 of 5

A student sets up \(\displaystyle V = \int_0^1 (e^x)^2\,dx\) for a solid with square cross sections and base under \(y = e^x\) from \(x = 0\) to \(x = 1\). What is the volume?