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.
Circle: Area
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
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.