The Timpson Lyceum · Calculus 1 (2024)

Volumes Using Cross-Sections and Cylindrical Shells - Lecture 17 (Calculus 1)

Overview

This lecture opens the study of applications of integration by reframing the integral not as "area" but as an infinite sum of infinitely small pieces. With that mindset, volume becomes the integral of infinitesimally thin slices. We develop four ways to compute volumes of solids: the general method of slicing (integrate a known cross-sectional area), the disk method and washer method (for solids of revolution whose slices are circles or rings), and the cylindrical-shell method (for solids of revolution built from nested thin tubes). The central skill is the setup: choosing a slicing direction, expressing the cross-section as a function of one variable, and reading off the correct bounds. Once the integral is set up, the integration itself is routine.

Key Concepts

Volume as a sum of slices

• A solid's volume is the integral of its infinitesimal slices: $V = \int dV$.
• Slice the solid perpendicular to one axis, find the cross-sectional area at position $x$ (call it $A(x)$), give it thickness $dx$, and sum: $V = \int_a^b A(x)\,dx$
• The slicing variable is arbitrary — you may instead slice in $y$ and use $V = \int_c^d A(y)\,dy$. The bounds are simply where you start and stop taking slices.

Method of slicing (known cross-sections)

• Works whenever you can determine $A(x)$ for every slice, even when the cross-section is a square, triangle, or other non-circular shape.
• Procedure: choose a coordinate system, draw a representative slice, find its area as a function of the slicing variable, then integrate.

Disk method (solid of revolution, no gap)

• When a region touching the axis of revolution is rotated, each slice is a solid disk of radius $R$, area $\pi R^2$.
• If the region lies between a curve and the axis and is revolved about that axis: $V = \pi \int_a^b [R(x)]^2\,dx$
• The radius $R$ is the distance from the axis to the curve. Slice in the direction you are spinning so the cross-sections are circles.

Washer method (solid of revolution, with gap)

• When the region does not touch the axis of revolution, each slice is an annulus (washer) with an outer radius $R$ and inner radius $r$, area $\pi(R^2 - r^2)$. $V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx$
• $R$ = distance from axis to the farther curve; $r$ = distance from axis to the nearer curve. Each radius is computed as (far edge) − (near edge).

Cylindrical-shell method

• Slice parallel to the axis of revolution; each slice sweeps out a thin hollow tube (shell). Unroll a shell into a flat rectangle of height $h$, length equal to the circumference $2\pi r$, and thickness $dx$ (or $dy$), giving shell volume $2\pi r\,h\,dx$. $V = 2\pi \int_a^b r(x)\,h(x)\,dx$
• $r$ = distance from the axis to the shell; $h$ = height of the region at that location.
• Useful when revolving about a vertical axis but the curve is given as $y = f(x)$ (avoids solving for $x$).

Common formulas recovered

• Cone/pyramid: $V = \tfrac13(\text{base area})(\text{height})$.
• Sphere of radius $a$: $V = \tfrac43 \pi a^3$ (derived by revolving a semicircle).

Choosing radii correctly

• A distance is always endpoint − start point. When revolving about a shifted axis such as $x = -1$, the shell radius is $x - (-1) = x + 1$.

Worked Example

Revolve the region under $y = \sqrt{x}$ from $x = 0$ to $x = 4$ about the $x$-axis. Each slice is a disk of radius $R = y = \sqrt{x}$, so the disk method gives:

The same solid can be found with cylindrical shells (slicing in $y$, radius $r = y$, height $h = 4 - x = 4 - y^2$), giving $V = 2\pi \int_0^2 y\,(4 - y^2)\,dy = 8\pi$ — the same answer, confirming the methods agree.

Summary

• Think of any volume as $V = \int dV$: pick a slicing direction, find the cross-sectional area as a function of one variable, multiply by the infinitesimal thickness, and integrate.
• The method of slicing handles any solid whose cross-sectional area $A(x)$ you can describe, including non-circular slices.
• The disk method, $V = \pi\int [R]^2\,dx$, applies to solids of revolution whose region touches the axis.
• The washer method, $V = \pi\int (R^2 - r^2)\,dx$, applies when a gap separates the region from the axis, so each slice is a ring.
• The shell method, $V = 2\pi\int r\,h\,dx$, slices parallel to the axis and sums thin tubes — often the easiest setup when revolving about a vertical axis.
• The hardest part is the setup, not the integration: draw the region, identify radii as (far edge) − (near edge), set the bounds to where slicing begins and ends, and choose whichever method keeps the algebra simplest.