Shell Integration

Shell Integration

How to find the volume of a rotated region by using the shell integration: formulas (x-axis, y-axis), proof, examples, and their solutions.

Formula 1: Rotated Around the y-axis

V = [inteval, from a to b, of (2*pi*x)*y dy]

The shell integration is an integration method
to find the volume of a rotated figure.

The 'shell' means a cylindrical shell.
It's a rotated 3D figure
whose cross-sectional area is a very thin ring.

If these cylindrical shells are filled under y = f(x)
from the inside (x = a) to the outside (x = b),
then V = ∫ab 2πxy dx.

The shell method is used
when the rotation axis (y-axis)
and the integrating direction (dx)
are different.

This is the big difference
between the shell integration and the disk integration.

Proof (Formula 1)

Shell Integration: Proof of the Formula (Rotated Around the y-axis)

Think of a cylindrical shell at x = x.

The radius of the base is r.
The height of the shell is y.
And the thickness of the shell is dx.

Let's cut this shell vertically
and unroll the shell.

Since dx is very small,
you can think this as a very thin 'right prism'.

The width is 2πx.

Circumference of a circle

The height is y.
And the depth is dx.

Then the volume of the right prism is
dV = 2πxy dx.

Volume of a right prism

Cylindrical shells are integrated
from x = a to x = b.

So V = ∫ab 2πxy dx.

Example 1

A region is bounded by y = x^4, x = 1, and the x-axis. If the region is rotated around the y-axis, find the volume of the rotated region.

First find the bounded region.

This gray colored region is the bounded region.

Draw the rotated figure
that is rotated around the y-axis.

And draw a cylindrical shell at x = x.

Before writing the integral,
sketch the details of the cylindrical shell.

The base's radius is x.
So the circumference is 2πx.

The height is y: x4.

The depth is dx.

And the integral interval is [0, 1].

So V = ∫01 2πxx4 dx.

Take 2π out from the integral.

And solve the integral.

Definite integration of polynomials

Example 2

A region is bounded by y = x^4, y = 1, and the y-axis. If the region is rotated around the y-axis, find the volume of the rotated region.

Previously, you've solved this example
by using the disk integration.

Let's solve the same example
by using the shell integration.

First find the bounded region.

This gray colored region is the bounded region.

Draw the rotated figure
that is rotated around the y-axis.

And draw a cylindrical shell at x = x.

Before writing the integral,
sketch the details of the cylindrical shell.

The base's radius is x.
So the circumference is 2πx.

The height is (1 - y), not y.
So the height is (1 - x4).

The depth is dx.

And the integral interval is [0, 1].

So V = ∫01 2πx⋅(1 - x4) dx.

Take 2π out from the integral.

And solve the integral.

Definite integration of polynomials

As you can see,
you can get the same answer: 2π/3.

Formula 2: Rotated Around the x-axis

V = [inteval, from a to b, of (2*pi*y)*x dx]

If the cylindrical shells are filled under x = g(y)
from the inside (y = a) to the outside (y = b),
then V = ∫ab 2πyx dy.

Example 3

A region is bounded by y = x^4, y = 1, and the y-axis. If the region is rotated around the x-axis, find the volume of the rotated region.

First find the bounded region.

This gray colored region is the bounded region.

The integration direction is y: dy.
So change y = x4 into x = y1/4.

Draw the rotated figure
that is rotated around the x-axis.

And draw a cylindrical shell at y = y.

Before writing the integral,
sketch the details of the cylindrical shell.

The base's radius is y.
So the circumference is 2πy.

The height is x: y1/4.

The depth is dy.

And the integral interval is [0, 1].

So V = ∫01 2πyy1/4 dy.

Take 2π out from the integral.

And solve the integral.

Definite integration of polynomials