# 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

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* = ∫_{a}^{b} 2*π**x*⋅*y* *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)

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 2*π**x*.

Circumference of a circle

The height is *y*.

And the depth is *dx*.

Then the volume of the right prism is*dV* = 2*π**x*⋅*y* *dx*.

Volume of a right prism

Cylindrical shells are integrated

from *x* = *a* to *x* = *b*.

So *V* = ∫_{a}^{b} 2*π**x*⋅*y* *dx*.

## Example 1

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 2*π**x*.

The height is *y*: *x*^{4}.

The depth is *dx*.

And the integral interval is [0, 1].

So *V* = ∫_{0}^{1} 2*π**x*⋅*x*^{4} *dx*.

Take 2*π* out from the integral.

And solve the integral.

Definite integration of polynomials

## Example 2

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 2*π**x*.

The height is (1 - *y*), not *y*.

So the height is (1 - *x*^{4}).

The depth is *dx*.

And the integral interval is [0, 1].

So *V* = ∫_{0}^{1} 2*π**x*⋅(1 - *x*^{4}) *dx*.

Take 2*π* out from the integral.

And solve the integral.

Definite integration of polynomials

As you can see,

you can get the same answer: 2*π*/3.

## Formula 2: Rotated Around the *x*-axis

If the cylindrical shells are filled under *x* = *g*(*y*)

from the inside (*y* = *a*) to the outside (*y* = *b*),

then *V* = ∫_{a}^{b} 2*π**y*⋅*x* *dy*.

## Example 3

First find the bounded region.

This gray colored region is the bounded region.

The integration direction is *y*: *dy*.

So change *y* = *x*^{4} into *x* = *y*^{1/4}.

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 2*π**y*.

The height is *x*: *y*^{1/4}.

The depth is *dy*.

And the integral interval is [0, 1].

So *V* = ∫_{0}^{1} 2*π**y*⋅*y*^{1/4} *dy*.

Take 2*π* out from the integral.

And solve the integral.

Definite integration of polynomials