What is the volume of a sphere when the radius is 9 m?

Answer 1

You can work this out if you know that the area of a circle is equal to pi times the square of it's radius, and that the equation for a circle is x2 + y2 = r2. Using a little bit of calculus, we can merge those two equations and say:

V =R∫-Rπ ⋅ y(x)2 dx

∴V =R∫-Rπ ((R² - x²)1/2 )2 dx

We can then take that integral to get the volume:

V = πR∫-R(R² - x²) dx

V = πR∫-RR² dx-πR∫-Rx² dx

V = πR2R∫-Rdx-πR∫-Rx2 dx

V = πR2(x)R

-R

-π(x3

3

)R

-R

V = πR2((R-(-R))-π

3

(R3 -(-R3 ))

V = πR3 + πR3 - πR3 /3 - πR3 /3

V = (2 - 2/3)πR3

V =4

3

πR3

We are told that the radius, R is equal to 9 meters, so the volume is equal to 4/3π93, or 972π.

Of course, most of us are just given the equation V=4/3πr3 to start with, but where's the fun in that?