Question

How to prove formula for volume of a sphere ?

Answer 1

The volume of radius is
`(4)/(3)` × π × radius³ Proved

Step-by-step explanation:

Given as :

We know that volume of sphere is v =
`(4)/(3)` × π × radius³

Or, v =
`(4)/(3)` × π × r³

Let prove the volume of sphere

So, From the figure of sphere

At the height of z , there is shaded disk with radius x

Let Find the area of triangle with side x , z , r

From Pythagorean theorem

x² + z² = r²

Or, x² = r² - z²

Or, x =
`\sqrt{r^(2)-z^(2) }`

Now, Area of shaded disk = Area = π × x²

Where x is the radius of disk

Or, Area of shaded disk = π × (
`\sqrt{r^(2)-z^(2) }`) ²

∴ Area of shaded disk = π × (r² - z²)

Again

If we calculate the area of all horizontal disk, we can get the volume of sphere

So, we simply integrate the area of all disk from - r to + r

i.e volume =
`\int_(-r)^(r) \Pi(r^(2)-z^(2) )dz`

Or, v =
`\int_(-r)^(r) \Pi r^(2)dz` -
`\int_(-r)^(r) \Pi z^(2)dz`

Or, v = π r² (r + r) - π
`(r^(3) -(-r)^(3)))/(3)`

Or, v = π r² (r + r) - π
`(2r^(3))/(3)`

Or, v = 2πr³ - π
`(2r^(3))/(3)`

Or, v = 2πr³ (
`(3-1)/(3)`)

Or, v = 2πr³ ×
`(2)/(3)`

∴ v =
`(4)/(3)` × π × r³

Hence, The volume of radius is
`(4)/(3)` × π × radius³ Proved . Answer