Volume and Arc Length Calculation with Integrals

We have seen that the integral can be used to calculate areas. We can also use the integral to find the length of a curve or even the volume of the shape created by rotating the graph of a function about the $x$ -axis.

Table of Contents

Solid of Revolution

Imagine sticking a straight metal skewer directly along the $x$ -axis and spinning the paper at a full 360-degree rotation. Take a function $f(x)$ from $x=a$ to $x=b$ . Using the metal skewer, we spin the paper and the flat 2D shape of the graph of $f(x)$ starts spinning, it carves out a solid, 3D object in space. The straight line along the $x$ -axis stays completely still, forming the solid core centre of the object.

The curve $f(x)$ spins through the air, forming the smooth outer crust/shell of the object.

Theorem (Volume of a Solid of Revolution – Disk Method): Let $f(x)$ be a continuous and non-negative function on the closed interval $[a, b]$ . When the region enclosed under the curve $y = f(x)$ and the $x$ -axis is rotated $360^\circ$ around the $x$ -axis, the volume $V$ of the resulting three-dimensional solid is given by:

$V = \pi \int_a^b \big[f(x)\big]^2 \, dx$

To use this formula properly, you should work in steps. First write the formula, then replace $f(x)$ and square it. Then, calculate the integral. If you are allowed a calculator which can do these integral calculations by itself, you can simply input $[f(x)]^2$ without expanding or simplifying your expression.

Example: Find the volume of the solid generated by revolving the region bounded by the curve $f(x) = x^2$ , the $x$ -axis, $x = 1$ , and $x = 4$ around the $x$ -axis.

Arc Length of a Function

Given some function $f(x)$ from $x=a$ to $x=b$ we can use integrals to calculate the length of the curve that is the graph of $f(x)$ . This is done as follows:

Theorem (Arc Length of a Function): Let $f(x)$ be a smooth function such that its first derivative $f'(x)$ is continuous on the closed interval $[a, b]$ . The arc length $S$ of the curve $y = f(x)$ from $x = a$ to $x = b$ is given by the definite integral:

$S = \int_{a}^{b} \sqrt{1 + \left[f'(x)\right]^2} \, dx$

Here we are also required to work in steps. First find the derivative $f'(x)$ . Then replace into the formula and simplify if necessary. If the resulting expression is simple enough, you may be able to find the integral without a technological tool. Let us see an example of how we apply this formula.

Example: Set up the definite integral required to calculate the exact arc length $S$ of the curve $y = x^2$ on the closed interval $[0, 1]$ . Then, use a calculator to approximate the total length to $3$ decimal places.

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Exercises

Calculate the volume of the solid generated by rotating the region bounded by the given curves around the $x$ -axis over the specified intervals.

1. $f(x) = \sqrt{x}$ , the $x$ -axis, from $x = 0$ to $x = 4$ .

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2. $f(x) = x^3$ , the $x$ -axis, from $x = 1$ to $x = 2$ .

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3. $f(x) = \frac{1}{x}$ , the $x$ -axis, from $x = 1$ to $x = e$ .

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Prove the classical geometric volume formula for a cone, $V = \frac{1}{3}\pi r^2 h$ , using the disk integration method.

Consider a straight line starting at the point $(0,r)$ on the $y$ -axis and slanting down to hit the $x$ -axis at the point $(h,0)$ . When the area beneath this linear profile line is spun $360^\circ$ around the $x$ -axis, it carves out a perfect cone with base radius $r$ and height $h$ . Refer to the graph below:

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Steps to Complete the Proof:

Find the equation of the line as a function of $r$ and $h$ .
Use the formula for the solid of revolution volume with boundaries: $a=0$ and $b=h$ .
Apply the Newton-Leibniz formula.

Prove the classical geometric volume formula for a cylinder, $V = \pi r^2 h$ , by applying the volume formula to the constant function $f(x)=r$ from $x=0$ to $x=h$ .

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For each of the following simple polynomial functions on their specified closed intervals, find the exact derivative expression, set up the definite integral for arc length, and then use your graphic or scientific calculator to compute the numerical result. Round all final answers to $3$ decimal places.

Function 1:
$f(x) = \frac{1}{3}x^3$
on $[0, 2]$

Function 2:
$g(x) = 2x - x^2$
on $[0, 2]$

Function 3:
$h(x) = x^4$
on $[0, 1]$

Function 4:
$j(x) = \frac{1}{2}x^2 + 3$
on $[1, 3]$

Differentiate the polynomial function to determine $y'$ .
Set up the arc length definite integral expression using the formula $S = \int_{a}^{b} \sqrt{1 + (y')^2} \, dx$ .
Evaluate the integral with a calculator.