Volume

Volumes Using Integrals

In this chapter we use definite integrals to find volumes of three–dimensional solids. The key idea is always the same:

• Slice the solid into very thin pieces whose volumes are easy to approximate.
• Add (integrate) the volumes of these slices.

Different kinds of solids suggest different kinds of slices, which lead to different standard methods.

We assume you are already familiar with definite integrals, areas, and the idea of approximating by Riemann sums from earlier chapters.

1. Solids of Revolution

A very common situation is when a region in the plane is rotated around an axis to form a solid. These are called solids of revolution.

You typically start with:

• A region in the $xy$–plane, usually bounded by one or more graphs, and
• A line (axis) about which that region is rotated.

The basic strategies are:

• Use cross-sections that are perpendicular to the axis (disk/washer method), or
• Use cross-sections that are parallel to the axis (shell method).

2. Disk Method

Use the disk method when rotating an area around an axis and the cross-sections perpendicular to that axis are solid disks (no holes).

2.1 Basic idea

Suppose a region between a curve and an axis of rotation is revolved around that axis.

• Think of slicing the solid by planes perpendicular to the axis.
• Each slice is a thin disk (a very flat cylinder).
• If the radius of the disk at position $x$ is $R(x)$ and the thickness is $dx$, then the volume of that slice is approximately
  $$dV \approx \pi [R(x)]^2\,dx.$$

Adding up all such slices from $x=a$ to $x=b$:

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

2.2 Rotating around the $x$–axis

If the region under $y=f(x)$, above the $x$–axis, from $x=a$ to $x=b$ is rotated around the $x$–axis, then at position $x$:

• The radius of the disk is $R(x)=f(x)$ (distance from the axis $y=0$ to the curve $y=f(x)$).
• Volume is
  $$
  V = \int_a^b \pi [f(x)]^2\,dx.
  $$

If the region is between the $x$–axis and the curve $y=f(x)\ge 0$, this gives a solid with no “hole” along the axis, so it is a pure disk method.

2.3 Rotating around the $y$–axis (using $y$ as variable)

If the region is described more naturally with $x$ as a function of $y$, say $x=g(y)$, and it is rotated around the $y$–axis, then:

• For a slice at height $y$, the radius is $R(y)=g(y)$.
• The thickness is $dy$.
• Volume:
  $$
  V = \int_c^d \pi [g(y)]^2\,dy,
  $$
  where $y$ runs from $c$ to $d$ over the region.

3. Washer Method

The washer method is an extension of the disk method when there is a hole along the axis, so the cross-section is a “washer” (a disk with a smaller disk removed).

3.1 Basic idea

If at a given $x$:

• The outer radius (distance to the outer boundary of the region) is $R(x)$.
• The inner radius (distance to the inner boundary, forming the hole) is $r(x)$.

Then the cross-sectional area is
$$
A(x) = \pi [R(x)]^2 - \pi [r(x)]^2.
$$

The volume is
$$
V = \int_a^b \big(\pi [R(x)]^2 - \pi [r(x)]^2\big)\,dx.
$$

3.2 Typical setup

Common cases:

• Region between two curves $y=f(x)$ (upper) and $y=g(x)$ (lower) revolved around the $x$–axis:
• Outer radius: $R(x)=f(x)$.
• Inner radius: $r(x)=g(x)$.
• Volume:
  $$
  V = \int_a^b \pi\Big(f(x)^2 - g(x)^2\Big)\,dx.
  $$
• If the axis of rotation is not one of the coordinate axes (e.g. $y=k$), you adjust radii to be distances to that line. For example, if rotating around $y=k$ and the curves are $y=f(x)$ (farther from $y=k$) and $y=g(x)$ (closer to $y=k$), then
  $$
  R(x) = |f(x) - k|,\quad r(x) = |g(x) - k|.
  $$

Similar formulas apply when integrating with respect to $y$ (i.e. using $dy$ slices and functions $x$ in terms of $y$).

4. Shell Method

The shell method uses slices parallel to the axis of rotation. Instead of disks or washers, you get thin cylindrical shells.

This method is often easier when:

• The functions are given as $y$ in terms of $x$ and you are rotating around a vertical line, or
• The functions are given as $x$ in terms of $y$ and you are rotating around a horizontal line.

4.1 Volume of a cylindrical shell

Imagine a thin cylindrical shell with:

• Radius $r$,
• Height $h$,
• Thickness $\Delta r$ (very small).

The lateral surface area is approximately $2\pi r h$, so the shell’s volume is
$$
\Delta V \approx 2\pi r h\,\Delta r.
$$

In the integral, $\Delta r$ becomes $dr$, and $r$ and $h$ become functions (like $x$ or $y$).

4.2 Rotating around the $y$–axis (using $x$–slices)

Suppose a region is bounded by curves $y=f(x)$ and $y=g(x)$ (with $f(x)\ge g(x)$) on $[a,b]$ and is revolved around the $y$–axis.

• Take a vertical slice at position $x$; its height is $h(x) = f(x) - g(x)$.
• Its distance to the axis $x=0$ (the $y$–axis) is $r(x)=x$.
• When rotated, this slice forms a cylindrical shell.

So
$$
V = \int_a^b 2\pi x\,[f(x)-g(x)]\,dx.
$$

If the axis is the vertical line $x=k$ rather than $x=0$, then
$$
r(x) = |x - k|.
$$

4.3 Rotating around the $x$–axis (using $y$–slices)

If you use horizontal slices and rotate around the $x$–axis:

• Radius is $r(y)$ = distance from the slice at height $y$ to the axis $y=0$, so $r(y)=y$ (or $|y-k|$ for $y=k$).
• Height is $h(y)$ = horizontal length of the region at that $y$ (difference of $x$–values of right and left boundaries).
• Volume:
  $$
  V = \int_c^d 2\pi r(y)\,h(y)\,dy.
  $$

5. Choosing Between Disk/Washer and Shell Methods

Often both methods are possible, but one is more convenient. Some guidelines:

• Use disk/washer when cross-sections perpendicular to the axis are easy to describe:
• If revolving around the $x$–axis, and the region is described as $y=f(x)$ on $[a,b]$, then disks or washers with $dx$ are natural.
• Revolving around the $y$–axis with region described as $x=g(y)$ suggests disks/washers with $dy$.
• Use shells when cross-sections parallel to the axis are simpler:
• If revolving around the $y$–axis and the region is given by $y=f(x)$, vertical shells with $dx$ are often easier.
• If revolving around the $x$–axis and the region is given by $x=g(y)$, horizontal shells with $dy$ are often easier.

In practice, try sketching the region and the axis of rotation. Then:

1. Draw a typical slice either perpendicular or parallel to the axis.
2. Ask: Is it easier to express the dimensions of that slice using $x$ or $y$?
3. Choose the method that leads to a simpler integrand and easier limits.

6. Volumes by Known Cross-Sections

Not all volume problems involve revolution. Sometimes the solid is defined by a base region and a description of the shape of cross-sections.

6.1 Basic idea

You are given:

• A base region in the plane (for example, a region $R$ in the $xy$–plane), and
• A rule describing the shape and size of cross-sections perpendicular to an axis (e.g. “Every cross-section perpendicular to the $x$–axis is a square”).

Let $A(x)$ be the area of the cross-section at position $x$. Then
$$
V = \int_a^b A(x)\,dx.
$$

Similarly, if cross-sections are perpendicular to the $y$–axis, you use
$$
V = \int_c^d A(y)\,dy.
$$

The main work is to express $A(x)$ or $A(y)$ in terms of the coordinate.

6.2 Common cross-sectional shapes

If the cross-section shape is specified, you use its usual area formula.

1. Squares: side length $s(x)$
   $$
   A(x) = [s(x)]^2.
   $$
2. Rectangles: base $b(x)$, height $h(x)$
   $$
   A(x) = b(x)\cdot h(x).
   $$
3. Semicircles: diameter $d(x)$
• Radius $r(x) = \dfrac{d(x)}{2}$.
• Full circle area is $\pi r^2$, so semicircle area:
  $$
  A(x) = \frac{1}{2}\pi [r(x)]^2 = \frac{1}{2}\pi \left(\frac{d(x)}{2}\right)^2.
  $$
4. Equilateral triangles: side length $s(x)$
• Area of equilateral triangle with side $s$ is $\dfrac{\sqrt{3}}{4}s^2$, so
  $$
  A(x) = \frac{\sqrt{3}}{4}[s(x)]^2.
  $$

In many problems, the side or diameter is the distance between two curves, such as $s(x) = f(x)-g(x)$.

6.3 Setting up the integral

Steps:

1. Graph the base region if possible.
2. Decide the direction of slicing (perpendicular to the $x$–axis or to the $y$–axis).
3. Find the length(s) that define your shape (e.g., side length, radius, diameter) in terms of $x$ or $y$ from the given curves.
4. Write the area formula $A(x)$ or $A(y)$.
5. Determine the integration limits from the base region.
6. Integrate $A(x)$ or $A(y)$ over those limits to get the volume.

7. Summary of Key Volume Formulas

All these formulas come from the same principle: volume $\approx$ (cross-sectional area)$\times$(thickness), and then take a limit to get an integral.

For convenience, here are the most used templates:

1. Disk method (about an axis, solid cross-sections):
   $$
   V = \int_a^b \pi [R(x)]^2\,dx
   \quad\text{or}\quad
   V = \int_c^d \pi [R(y)]^2\,dy.
   $$
2. Washer method (about an axis, hollow inside):
   $$
   V = \int_a^b \pi\big([R(x)]^2 - [r(x)]^2\big)\,dx
   \quad\text{or}\quad
   V = \int_c^d \pi\big([R(y)]^2 - [r(y)]^2\big)\,dy.
   $$
3. Shell method:
• About the $y$–axis (vertical axis, use $x$–slices):
  $$
  V = \int_a^b 2\pi x\,h(x)\,dx
  \quad\text{(or }2\pi |x-k|\,h(x)\text{ about }x=k\text{)}.
  $$
• About the $x$–axis (horizontal axis, use $y$–slices):
  $$
  V = \int_c^d 2\pi y\,h(y)\,dy
  \quad\text{(or }2\pi |y-k|\,h(y)\text{ about }y=k\text{)}.
  $$
4. Known cross-sections:
   $$
   V = \int_a^b A(x)\,dx
   \quad\text{or}\quad
   V = \int_c^d A(y)\,dy.
   $$

Later practice problems will help you choose the appropriate method and translate geometric descriptions into integral expressions for volume.