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How to calculate volume

Rotating about the y axis

In order to calculate the volume of something, first we represent it with a line, y = something (without the axis cutting it):
A graph of a line
and rotate it around the y axis which makes a solid. If we chop it up in to small cylinders...

then we can approximate it by adding up these cylinders. The volume of a cylinder is πr2h so since this is on its side, Δx represents the height of a small cylinder. The radius will be the value of the y value so the volume of one small cylinder is πy2(Δx).
If we could add up lots of cylinders, we could get an approximate answer:
Volume ≈  x=b
∑
x=a 		πy2(Δx)
We can make Δx smaller and smaller (Δx → 0) to get a result and we are left with:
Volume =  ∫ b πy2 dx
a

I shall present some examples later. Let's have another equation:

Rotating about the x axis

Again, we take a line y = something (without the x axis intersecting it):

The value of y cannot be 0 otherwise this would make a disc which has no volume.
We rotate the line around the x axis:

Now we have a similar idea of chopping it up in to bits. This time we want a thin cylinder:

To calculate its area, we multiply its cross-section which is y•(Δx) and multiply it with the circumference which is 2πx (which resembles the equation for the circumference of a circle, 2πr).
The volume of a thin cylinder is approximately y•(Δx)•2πx so if we add them all up, we have an approximate answer to the volume of the whole shape:
Volume ≈  x=b
∑
x=a 		2πxy•(Δx)
We can make Δx smaller and smaller (Δx → 0) to get a result and we are left with:
Volume =  ∫ b 2πxy dx
a

You can see some constants so I can put them outside of the integrand just to make calculations a bit easier:
Volume = 2π ∫ b xy dx
a


To summarize:
The volume of a shape is:
Volume = π ∫ b y2 dx (rotating around the y axis)
a
Volume = 2π ∫ b xy dx (rotating around the x axis)
a



The volume of a cone

To illustrate the above, let's calculate the volume of a cone.
We need a line equation. This is easy as it is a straight line (r is the radius of the base of the cone and h is the height):
y
  = x r
h

I shall rotate it about the y axis:

...and we have a cone.
Because I have rotated it around the y axis, I use:
Volume = π ∫ b y2 dx
a
I calculate the volume from 0 to h (h is the height of the cone)
Volume = π ∫ h (rx / h)2 dx
0
I can square it and take out the constants (which makes things easier)
Volume = π r2 ∫ h x2 dx
h2 0
This is very easy to calculate since the 0 removes a part of the equation and you get:
Volume = π r2
h2		 ( h3
3		 )  =  πr2h
3

The volume of part of a sphere

A similar example above is to first get an equation for half a circle, y = √(r2 - x2):


Now I am going to shift it to the right:
y = √(r2 - (x - r)2) = √(r2 - (x2 - 2xr + r2)) = √(2xr - x2)

Let's look at the sphere:

If I split the sphere in to many slices, I have lots of discs which all have an area of πr2
I am going to get my equation for half a circle and use it in place of the variable r (the radius):
π(√(2xr - x2))2 = π(2xr - x2)
All I need to do is integrate this from 0 to h
Volume of a section of a sphere = π ∫ h 2xr - x2 dx
0
= π ( h2r -  h3
3		 )

Volume of a section of a sphere = πh2 ( r -  h
3		 )


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