Question

How to prove formula for volume of a sphere ?

Answer 1

Answer:

The volume of radius is   $\frac{4}{3}$ × π × radius³  Proved

Explanation:

Given as :

We know that volume of sphere is v = $\frac{4}{3}$ × π × radius³

Or, v =  $\frac{4}{3}$ × 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 $\frac{r^{3} -(-r)^{3})}{3}$

Or, v = π r² (r + r) - π $\frac{2r^{3}}{3}$

Or, v = 2πr³ - π $\frac{2r^{3}}{3}$

Or, v =  2πr³ ($\frac{3-1}{3}$)

Or, v = 2πr³ × $\frac{2}{3}$

∴ v =   $\frac{4}{3}$ × π × r³

Hence, The volume of radius is   $\frac{4}{3}$ × π × radius³  Proved . Answer