Volume And Surface Area Of Solids Of Revolution 93

1. Core Concept Overview & Intuition

A solid of revolution is generated when a plane region is rotated about a fixed straight line, called the axis of revolution. When a curve $y=f(x)$ lying above the $x$-axis on an interval $[a,b]$ is rotated completely about the $x$-axis, the curve sweeps out a closed surface and the region beneath it sweeps out a solid. The study of such solids belongs to the applications of the definite integral, and it rests on two distinct but parallel constructions: one for the volume enclosed by the solid, and one for the area of its curved boundary surface.

The intuition for the volume formula is that of slicing. A thin vertical strip of the region, of width $\Delta x$ and height $y=f(x)$, when rotated about the $x$-axis, produces a thin circular disc of radius $y$ and thickness $\Delta x$. The volume of that disc is $\pi y^{2}\Delta x$. Summing the volumes of all such discs across $[a,b]$ and passing to the limit of vanishing width yields a definite integral. The solid is thus decomposed into an infinite stack of infinitesimally thin discs, and the total volume is the accumulated sum of their volumes.

The intuition for the surface area formula is subtler. The curved boundary of the solid is generated not by the vertical strip but by the arc of the curve itself. A short element of arc of length $\Delta s$, situated at distance $y$ from the axis, when rotated about the axis, sweeps out a narrow band whose area is approximately the circumference of the circle traced by the element multiplied by the arc length, that is $2\pi y\,\Delta s$. Summing these bands and passing to the limit produces a definite integral in which the arc-length element $ds$ appears. This is why the surface-area integrand contains the derivative factor $\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}$, whereas the volume integrand does not.

The essential contrast to retain is therefore the following. Volume accumulates cross-sectional disc areas $\pi y^{2}$ along the axis, while surface area accumulates circumferential band areas $2\pi y$ weighted by the arc-length element $ds$. Both are definite integrals over the same interval $[a,b]$, and both require the curve to be smooth enough for the relevant integrand to exist.

2. Mathematical Formalization & Core Definitions

> Definition (Solid of Revolution). Let $y=f(x)$ be a continuous, non-negative function on a closed interval $[a,b]$. The region bounded by the curve $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$, when rotated completely about the $x$-axis, generates a solid called a solid of revolution. The curve $y=f(x)$ generates the curved surface of the solid, and the axis of rotation is the $x$-axis.

> Definition (Element of Arc Length). For a curve $y=f(x)$ with continuous derivative, the element of arc length, denoted $ds$, is the infinitesimal length of a short piece of the curve. It is expressed in terms of $x$ by

$$
ds=\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx.
$$

> The quantity $ds$ is the infinitesimal limit of the chord length joining two neighbouring points of the curve, and it is the same element that underlies the length of an arc of a curve.

> Definition (Volume of a Solid of Revolution). The volume $V$ of the solid generated by rotating the region under $y=f(x)$ on $[a,b]$ about the $x$-axis is the limit of the sum of the volumes of the thin discs of radius $y$ and thickness $\Delta x$ obtained by slicing the solid perpendicular to the axis. In integral form this limit is written

$$
V=\int_{a}^{b}\pi y^{2}\,dx.
$$

> Definition (Surface Area of a Solid of Revolution). The surface area $S$ of the curved surface generated by rotating the arc $y=f(x)$, $a\le x\le b$, about the $x$-axis is the limit of the sum of the areas of the narrow bands swept out by the elements of arc. In integral form this limit is written

$$
S=\int_{a}^{b}2\pi y\,ds=\int_{a}^{b}2\pi y\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx.
$$

The symbols are interpreted as follows. The variable $x$ is the coordinate along the axis of revolution, measured in the same linear units as $y$. The function $y=f(x)$ is the distance from the axis of revolution to the curve, so it plays the role of the radius of the disc in the volume formula and of the radius of the band in the surface-area formula. The constant $\pi$ is the ratio of the circumference of a circle to its diameter. The limits $a$ and $b$ are the endpoints of the interval of revolution, with $a<b$. The quantity $V$ is measured in cubic units and $S$ in square units, consistent with the linear units of $x$ and $y$.

Both formulae presuppose that $f$ is non-negative on $[a,b]$, so that $y$ is a genuine distance from the axis, and that $f$ is sufficiently smooth for the integrands to be integrable. The volume integrand $\pi y^{2}$ requires only that $f$ be continuous, whereas the surface-area integrand requires that $f$ possess a continuous derivative, because the factor $\sqrt{1+(dy/dx)^{2}}$ must be defined throughout the interval.