Disc Integration

How to use the disc integration to find the volume of a rotated figure about the axis (x-axis and y-axis): formula, 3 examples, and their solutions.

Formula

Formula: Rotated about the x-axis

To find the volume of a rotated figure
about the x-axis,

add up the cross sectional slices,
which are discs.

Volume from Its Slices

The cross section is a circle
whose radius is y.

So the cross sectional area is πy2.

Area of a Circle

Then volume of each slice is
πy2⋅dx = πy2 dx.

So the volume of the rotated figure
about the x-axis is
V = ∫ab πy2 dx.

Formula: Rotated about the y-axis

By the same way,
the volume of a rotated figure
about the y-axis is
V = ∫ab πx2 dy.

Example 1

Example

Solution

Draw y = x4.
Draw x = 1.

Then color the region
that is bounded by these two graphs and the x-axis.

Draw the rotated figure.

Draw a cross sectional disc.

The radius of the disc is y.
So write the radius y = x4.

x is from 0 to 1.
The radius of the disc is y = x4.
And the region is rotated about the x-axis.

Then V = ∫01 π⋅(x4)2 dx.

01 π⋅(x4)2 dx
= π∫01 x8 dx

Power of a Power

Solve the integral.

Definite Integral: How to Solve

The integral of x8 is
[1/9]x9.

Integral of a Polynomial

Put 1 and 0
into [1/9]x9.

[1/9]⋅19 = 1/9
-[1/9]⋅09 = -0

π(1/9 - 0) = π/9

So π/9 is the answer.

Example 2

Example

Solution

Draw y = x4.
Draw y = 1.

Then color the region
that is bounded by these two graphs and the y-axis.

Draw the rotated figure.

Draw a cross sectional disc.

The radius of the disc is x.
So write the radius x = y1/4.

y is from 0 to 1.
The radius of the disc is x = y1/4.
And the region is rotated about the y-axis.

Then V = ∫01 π⋅(y1/4)2 dy.

01 π⋅(y1/4)2 dy
= π∫01 y1/2 dy

Solve the integral.

The integral of y1/2 is
[2/3]y3/2.

Integral of a Radical

Put 1 and 0
into [2/3]y3/2.

[2/3]⋅13/2 = 2/3
-[2/3]⋅03/2 = -0

π(2/3 - 0) = 2π/3

So 2π/3 is the answer.

Example 3: Proof of the Volume of a Sphere Formula

Example

Recall that
if the radius of a sphere is r,
then the volume of the sphere is
[4/3]πr3.

Let's see the proof of this formula.

Solution

To make a sphere whose radius is r,
first draw a circle
whose radius is r.

The equation of the circle is
x2 + y2 = r2.

Rotate the circle about the x-axis.

Then the rotated figure is a sphere
whose radius is r.

Draw a cross sectional disc.

The radius of the disc is y.
So write y2 = r2 - x2.

x is from -r to r.
The radius of the disc is y
that satisfies y2 = r2 - x2.
And the region is rotated about the x-axis.

Then V = ∫-rr π⋅(r2 - x2) dx.

Take the constant π out from the integral.

The upper limit (r) and the lower limit (-r)
are the opposites.

r2 and -x2 are even functions.

So π∫-rr (r2 - x2) dx
= π⋅2∫0r (r2 - x2) dx.

Definite Integral: Odd and Even Functions

π⋅2 = 2π

Solve the integral.

The integral of r2 is
r2x.

The integral of -x2 is
-[1/3]x3.

Put r and 0
into r2x - [1/3]x3.

r2⋅r = r3

-(r2⋅0 - [1/3]⋅03) = -(0)

[r3 - [1/3]r3 - 0] = r3(1 - 1/3)

1 - 1/3 = 2/3

2π⋅r3⋅[2/3] = [4/3]πr3

So the volume of the sphere is [4/3]πr3.

This is the proof of the volume of a sphere formula.