In this video we will derive the volume formula
for a sphere using integration,
where the volume of the sphere equals
four thirds pi r cubed.
To begin, one way to create a sphere
would be to graph this circle x squared
plus y squared equals r squared
on the x y coordinate plane and then
rotate the circle about the x axis.
After the rotation we have a sphere
with radius r.
And now to determine the volume
of the sphere we can think of
making vertical slices of this sphere
which vertical slice would be a disk
or right circular cylinder as pictured here.
Remember the volume of a right circular cylinder
is pi r squared h and for this disk
a right circular cylinder
notice how the radius is y and the height
would be from a small change in x
represented by delta x or differential x.
So we can say the volume of one disk
or one slice of this sphere equals
pi y squared times differential x.
From here as the number of slices
approach infinity or the number
of disks approach infinity
the sum of the volumes of the disks
or the right circular cylinders
will approach the volume of the sphere.
And now let's introduce some calculus notation.
Let's first focus on this sum here.
We've got the sum from i equals one to n
of pi y sub i squared times delta x.
Well pi y sub i squared times delta x
represents the volume of one disk
or right circular cylinder and therefore
this sum represents the sum of n disks.
And again as the number of disks approach infinity
the sum of the volumes of the disks
approach the volume of the sphere.
And we know the limit of this sum
is equal to the definite integral.
from negative r to r of pi y squared differential x.
Notice here we're integrating from negative r to r
all along the x axis.
From here though let's make two changes.
Instead of integrating from negative r to r
let's integrate from zero to r
then multiply the integral by two.
Also because of the differential x
we know we have to integrate with respect to x
and therefore we need to perform
a substitution for y squared using the
equation x squared plus y squared equals r squared.
Solving for y squared we have y squared
equals r squared minus x squared.
So for this next line we can say
the volume equals two times integral
from zero to r, this changes from integrating
from zero to r rather than negative r to r.
And then performing a substitution for y squared
we have r squared minus x squared here.
So the next step we can factor out pi
which gives us the volume is equal to two pi
times integral from zero to r of r squared
minus x squared differential x.
So the next step we're going to get with respect to x
treating r squared as a constant.
The anti derivative is r squared times x
minus x cubed divided by three.
Again we have r squared x here because
we treat r squared as a constant
From here we need to find big F of b minus big F of a
or in this case big F of r minus big F of zero.
To begin we substitute r for x which gives us r
squared times r giving us r cubed minus
substituting r for x here it gives us r
cubed divided by three minus now we find big F of zero
by substituting zero for x which gives us
r squared times zero minus zero cubed
divided by three.
Simplifying this is zero and so is this.
Here we have one r cubed minus one third r cubed
which gives us two thirds r cubed
giving us two pi times two thirds r cubed
which does give us four thirds pi r cubed
which is the volume formula for a sphere.
I hope you found this helpful.