Volumes with Cross Sections

When we slice a three-dimensional solid perpendicular to an axis, we get a two-dimensional shape called a cross section. If we know how the area of that cross section changes as we move along the axis, we can recover the entire volume of the solid by integration — even if the solid has no simple geometric formula. This is the fundamental idea behind the method of volumes with known cross sections. The technique generalizes the familiar disk and washer methods: instead of insisting that each cross section is a circle or annulus, we allow the cross sections to be squares, equilateral triangles, rectangles, semicircles, or any shape whose area we can express as a function of a single variable. The integral then 'adds up' the infinitely thin slices, each of volume A(x) dx, to give the total volume. This method appears throughout calculus, physics, and engineering whenever a shape is described by its profile rather than a closed-form equation. Architects use it to compute the volume of unusual arches; physicists use it to find the mass of objects with varying density cross sections. Mastering this concept deepens your understanding of the definite integral as an accumulation tool.

Key takeaways
  • Identify the axis of slicing
  • Find the edge length (or relevant dimension) of each slice
  • Write the area of one cross section A(x)
  • Integrate A(x) over the interval
  • Evaluate and interpret

Identify the axis of slicing

Decide whether to slice perpendicular to the x-axis or y-axis. The slices should be parallel to the given cross-sectional shape. For most problems, slicing perpendicular to the x-axis (vertical slices) is natural when the base region is described by functions of x.

Find the edge length (or relevant dimension) of each slice

At position x, the cross section has some characteristic length. If the base of the solid lies between two curves y = f(x) and y = g(x) with f(x) ≥ g(x), then the edge length (or diameter, or base) of the cross section at x is s(x) = f(x) − g(x). This is the 'width' of the base region at that x-value.

Write the area of one cross section A(x)

Use the geometry of the specified shape. For a square cross section with side s: A(x) = [s(x)]². For an equilateral triangle with side s: A(x) = (√3/4)[s(x)]². For a semicircle with diameter s: A(x) = (π/8)[s(x)]². For a rectangle with one side s and height h: A(x) = h · s(x). Always express A entirely in terms of x.

Integrate A(x) over the interval

Determine the limits of integration [a, b] from the base region (often intersection points of the two curves). Then compute V = ∫ from a to b of A(x) dx. Each infinitesimal slice contributes volume dV = A(x) dx, and the integral accumulates all of them.

Evaluate and interpret

Carry out the definite integral using antiderivatives or substitution as needed. The result is the exact volume in cubic units. Always check units: if lengths are in centimetres, the volume is in cm³. A rough estimate (base area × average height) can serve as a sanity check.

The Key Equation

V=∫abA(x) dxV = \int_{a}^{b} A(x)\, dx
  • V — Volume (cubic units (e.g. cm³)): The total three-dimensional volume of the solid.
  • A(x) — Cross-sectional area function (square units): The area of the slice taken perpendicular to the x-axis at position x. Its formula depends on the shape of the cross section.
  • dx — Infinitesimal thickness (units of length): The infinitely thin thickness of each slice; multiplied by A(x) it gives the infinitesimal volume dV = A(x) dx.
  • a,\, b — Limits of integration (units of length): The x-values at the two ends of the solid, typically found from intersection points of the boundary curves.

Worked Example

Problem. A solid has as its base the region bounded by y = x² and y = 4 (for −2 ≤ x ≤ 2). Cross sections taken perpendicular to the x-axis are squares. Find the volume of the solid.

V=51215 cubic unitsV = \dfrac{512}{15} \text{ cubic units}

Answer. The volume is 512/15 cubic units.

Figure
Diagram
Volumes with Cross Sectionsxyy = 4y = x²s(x)squarecross sectionBase region between the two curves (shaded blue)A(x) = [s(x)]² for square slices
Origins · 1 minBonaventura Cavalieri and how this was discovered
BC
Bonaventura Cavalieri
1598–1647 · Italian

Cavalieri developed the method of indivisibles (around 1635), an early form of integration in which a solid is thought of as composed of infinitely many parallel cross sections — exactly the idea underlying the volume-by-cross-sections technique. His 'Cavalieri's Principle' states that two solids with equal cross-sectional areas at every height have equal volumes, a result still cited in modern calculus courses. Cavalieri was a Jesuit monk and a student of Galileo. He wrote his major work Geometria Indivisibilibus while suffering from severe gout, completing much of it while bedridden.

  • Disk and Washer Method — The disk and washer method is a special case of volumes by cross sections where every cross section is circular; understanding both methods reveals the full power of the integral as an accumulation tool.
  • The Definite Integral — The volume formula V = ∫A(x)dx is a direct application of the definite integral as a net accumulation of infinitesimal quantities.
  • Shell Method for Volumes of Revolution — The shell method offers an alternative approach to finding volumes by summing cylindrical shells rather than cross-sectional slices, highlighting different strategies for the same type of problem.
  • Area Formulas for Plane Figures — Computing A(x) for squares, triangles, and semicircles requires fluency with 2D area formulas, which are the building blocks of every cross-section volume calculation.