Question

Derive the formula for the volume of a sphere?

Answer 1

`V = 4/3pi a^3`

Explanation:

Consider a 3-dimensional sphere of radius `a`:

In the `xy`-plane, we have a circle of radius `a`, with equation:

`x^2+y^2=a^2`

Then if we take an arbitrary `x`-coordinate, and take an infinitesimally thin vertical slice through the sphere we will have a circle of radius `sqrt(a^2-x^2)` (red), so the area of the red circle is:

`A = pir^2`
`\ \ \ = pi(sqrt(a^2-x^2))^2`
`\ \ \ = pi(a^2-x^2)`

We need to sum all of these infinitesimally thin vertical red slices as `x` varies from `-a` to `a`, hence the total volume is given:

`V = int_(-a)^(a) \ pi(a^2-x^2) \ dx`
`\ \ \ = pi \ int_(-a)^(a) \ a^2-x^2 \ dx`
`\ \ \ = pi \ [a^2x-x^3/3]_(-a)^(a)`
`\ \ \ = pi \ {(a^3-a^3/3) - (-a^3+a^3/3)}`
`\ \ \ = pi \ (a^3-a^3/3 +a^3-a^3/3)`
`\ \ \ = pi \ (2a^3-2/3a^3 )`
`\ \ \ = pi \ (4/3a^3)`
`\ \ \ = 4/3pi a^3`