6.3 Volumes of Revolution: Cylindrical Shells

Volumes of Revolution: Cylindrical Shells

Cylindrical shells volume calculation

The shell method works by slicing a region into thin vertical rectangles and then rotating each one around a vertical axis. When a rectangle rotates, it sweeps out a hollow cylinder (a "shell"), and you sum up the volumes of all those shells using integration.

Each thin shell has three measurements:

• $r$ (radius): the distance from the axis of rotation to the center of the shell
• $h$ (height): the height of the rectangle, determined by the function $f(x)$
• $\Delta x$ (thickness): the width of the thin rectangle

The volume of a single shell is approximately $2\pi r h \Delta x$. Think of it as "unrolling" the cylinder into a flat slab: its length is the circumference $2\pi r$, its height is $h$, and its thickness is $\Delta x$.

To get the exact total volume, you integrate over the interval $[a, b]$:

$V = \int_{a}^{b} 2\pi \, r \, h \, dx$

How you define $r$ and $h$ depends on the axis of rotation:

• Rotation around the y-axis: $r = x$ and $h = f(x)$
• Rotation around a vertical line $x = c$: $r = |x - c|$ and $h = f(x)$

Example: Rotate the region bounded by $y = x^2$ and $y = 4$ around the y-axis. The curves intersect where $x^2 = 4$, so $x = 0$ to $x = 2$ (taking the right side). Each shell has radius $r = x$ and height $h = 4 - x^2$, giving:

$V = \int_{0}^{2} 2\pi \, x(4 - x^2) \, dx$

Example: Rotate the region under $y = \sin(x)$ from $x = 0$ to $x = \pi$ around the line $x = \pi$. Here $r = \pi - x$ and $h = \sin(x)$:

$V = \int_{0}^{\pi} 2\pi(\pi - x)\sin(x) \, dx$

Method selection for revolution volumes

Picking between shells and disks/washers comes down to which method gives you a simpler integral. The guiding question is: does your slice run parallel or perpendicular to the axis of rotation?

• Use shells when the region is described by functions of $x$ and you're rotating around a vertical axis. Your slices are vertical rectangles that run parallel to the axis. This is especially helpful when the region is bounded by two $x$-functions (like $y = x^2$ and $y = x^3$), because the height of each shell is just the difference of the two functions.
• Use disks/washers when the region is described by functions of $x$ (or $y$) and you're rotating around a horizontal axis. Your slices are perpendicular to the axis and form circular cross-sections. For instance, rotating $y = \sqrt{x}$ around the x-axis is straightforward with washers.

Sometimes both methods work, but one produces a much cleaner integral. Before committing, sketch the region and the axis of rotation, then ask yourself:

1. Which variable will I integrate with respect to?
2. Will I need to split the integral into multiple pieces?
3. Would I have to solve for $x$ in terms of $y$ (or vice versa)?

If shells let you avoid solving for a new variable or splitting the region, they're probably the better choice.

Example: Rotating the region under $y = e^x$ from $x = 0$ to $x = 1$ around the y-axis. Using washers here would require rewriting $x = \ln(y)$ and dealing with awkward bounds. Shells keep things clean: $V = \int_{0}^{1} 2\pi x \, e^x \, dx$.

Example: Rotating the region bounded by $y = x^2$ and $x = 1$ around the x-axis. The cross-sections perpendicular to the x-axis are simple washers, so disks/washers is the natural choice.

Off-axis rotation volume calculation

When the axis of rotation is a vertical line other than the y-axis, the setup is almost identical to the standard shell method. The only change is how you compute the radius.

Steps for off-axis shell problems:

1. Sketch the region and mark the axis of rotation $x = c$.

2. Identify the bounds $a$ and $b$ for the region along the x-axis.

3. For a shell at position $x$, compute the radius as $r = |x - c|$. If every $x$ in your interval is on the same side of the line $x = c$, you can drop the absolute value and just use $r = x - c$ or $r = c - x$, whichever is positive.

4. Determine the height $h$ from the bounding function(s).

5. Integrate:

$V = \int_{a}^{b} 2\pi \, |x - c| \, h \, dx$

Example: Rotate the region bounded by $y = x^2$ and $y = 4$ around the line $x = 2$. The region runs from $x = 0$ to $x = 2$, so every shell is to the left of $x = 2$. The radius is $r = 2 - x$ (positive throughout), and the height is $h = 4 - x^2$:

$V = \int_{0}^{2} 2\pi(2 - x)(4 - x^2) \, dx$

Example: Rotate the region under $y = \cos(x)$ from $x = 0$ to $x = \pi/2$ around the line $x = \pi$. Every shell is to the left of $x = \pi$, so $r = \pi - x$:

$V = \int_{0}^{\pi/2} 2\pi(\pi - x)\cos(x) \, dx$

Fundamentals of Volumes of Revolution

A solid of revolution is the 3D shape you get by spinning a 2D region around an axis. Every method for finding its volume relies on the same core idea: slice the solid into pieces whose volumes you can calculate, then add them all up with a definite integral.

• A cross-section is the shape you see when you cut the solid perpendicular to the axis of rotation. For disks/washers, these cross-sections are circles or rings. For shells, you're instead summing cylindrical surfaces parallel to the axis.
• The definite integral acts as the summation tool. It takes infinitely many infinitesimally thin slices and totals their volumes into one exact number.
• The area of the original 2D region directly affects the volume. A larger region swept through the same rotation produces a larger solid.