Volumes by Shells

What You’ll Learn

In this lesson you’ll learn the shell method, an alternative to disks and washers for finding volumes of solids of revolution. It’s often the easier choice when rotating around the y-axis.

The Concept

Instead of slicing the solid into flat disks, the shell method wraps the solid in thin cylindrical shells (like nesting tubes). Each shell has a radius, a height, and a tiny thickness.

The formula

When rotating a region around the y-axis

$$V = 2\pi \int_a^b x \cdot h(x)\,dx$$

where x is the radius of the shell (distance from the y-axis) and h(x) is the height of the shell at that x value.

The idea: circumference times height times thickness = volume of one thin shell. Add them all up (integrate) to get the total volume.

When to use shells vs disks/washers

Use shells when the axis of rotation is parallel to the variable you’re integrating with respect to. Specifically:

• Rotating around the y-axis and your functions are in terms of x? Shells are usually easier.
• Rotating around the x-axis and your functions are in terms of y? Shells again.
• If you’d have to solve for the other variable or split into multiple integrals with disks, try shells instead.

Both methods give the same answer. Pick whichever makes the integral simpler.

Worked Example

Example 1: Region between y = x and y = x² rotated around the y-axis

Find the volume when the region between y = x and y = x² from x = 0 to x = 1 is rotated around the y-axis.

Drag to rotate · Scroll to zoom

This is the same solid from the washer lesson, but the shell method makes the setup much simpler. Each shell has radius x and height (x - x²).

$$V = 2\pi \int_0^1 x(x - x^2)\,dx = 2\pi \int_0^1 (x^2 - x^3)\,dx$$ $$= 2\pi\left[\frac{x^3}{3} - \frac{x^4}{4}\right]_0^1 = 2\pi\left(\frac{1}{3} - \frac{1}{4}\right) = 2\pi \cdot \frac{1}{12} = \frac{\pi}{6}$$

This is the same region from the washer lesson, but notice how much simpler the shell setup is. One integral, no solving for x in terms of y.

Example 2: y = sqrt(x) from x = 0 to x = 4 rotated around the y-axis

Drag to rotate · Scroll to zoom

The orange inner surface is the paraboloid from y = sqrt(x), and the blue outer cylinder is the boundary at x = 4. The green profile curve shows y = sqrt(x). Each shell has radius x and height sqrt(x).

$$V = 2\pi \int_0^4 x \cdot \sqrt{x}\,dx = 2\pi \int_0^4 x^{3/2}\,dx$$ $$= 2\pi\left[\frac{x^{5/2}}{5/2}\right]_0^4 = 2\pi \cdot \frac{2}{5}\left[x^{5/2}\right]_0^4 = \frac{4\pi}{5} \cdot 4^{5/2} = \frac{4\pi}{5} \cdot 32 = \frac{128\pi}{5}$$

Example 3: Comparing shell and washer methods

For Example 1, the washer method around the y-axis would require solving y = x for x = y and y = x² for x = sqrt(y), then integrating

$$V = \pi \int_0^1 \left[(\sqrt{y})^2 - y^2\right]\,dy = \pi \int_0^1 (y - y^2)\,dy = \pi\left[\frac{y^2}{2} - \frac{y^3}{3}\right]_0^1 = \pi\left(\frac{1}{2} - \frac{1}{3}\right) = \frac{\pi}{6}$$

Same answer, but you had to invert both functions first. The shell method skipped that step entirely.

Real-World Application

The shell method shows up anywhere you need volumes of rotationally symmetric objects.

• In manufacturing, designing containers and machine parts where the profile is easier to describe from the side
• In engineering, calculating volumes of pipes, turbines, and axially symmetric components
• In game development, generating 3D objects by revolving a profile curve, especially when the profile is defined as y = f(x) and the rotation axis is vertical
• In physics, computing moments of inertia for cylindrical and conical objects

You’ve Got This

The shell method boils down to one formula: 2 pi times radius times height times dx. The radius is the distance from the axis (usually just x when rotating around the y-axis), and the height is the function value (or the gap between two functions). If you can identify those two things, the integral writes itself. Try both methods on the same problem a few times to build intuition for which one is simpler.

Quiz

The shell method formula for rotation around the y-axis is

• A.pi times the integral of R squared dx
• B.2 pi times the integral of x times h(x) dx
• C.pi times the integral of (outer squared minus inner squared) dx
• D.The integral of f(x) dx

The volume from rotating the region between y = x and y = x² (from 0 to 1) around the y-axis using shells is

• A.pi/3
• B.2 pi/15
• C.pi/6
• D.pi/4

The shell method is often preferred over washers when

• A.The region touches the axis of rotation
• B.You'd have to invert the function to use washers
• C.The function is a constant
• D.The limits are 0 to infinity

In the shell method, the 'radius' of each shell when rotating around the y-axis is

• A.The y-value
• B.The height of the function
• C.The x-value (distance from the y-axis)
• D.Always 1

The shell and washer methods applied to the same solid give

• A.Different volumes
• B.The same volume
• C.The shell method always gives a larger answer
• D.Results that can't be compared