Another math question. How do I use integration to find the volume of a yellow plastic ball.?

Question

This ball in 3.0 inches in height. The circumference is 10.0 inches. Therefore, the radius is 5 inches. I know that V=(pi) r squared) (height). My question is how do I integrate this formula?

Answer 1

The radius is half the diameter, not half the circumference, so the radius is 1.5".

The easiest way to do it is a triple integral using spherical coordinates. I will derive the general-form equation for volume of a sphere, and then we can plug in the radius of 1.5" to find the specific result.

In 3-D spherical coordinates, integrate half a sphere and double the result to avoid the sinusoid cancelling itself in a full rotation...

2 S[θ=0:180°] S[φ=-90:90°] S[r=0:R] r² sinθ dr dφ dθ
2 S[θ=0:180°] S[φ=-90:90°] (R³/3) sinθ dφ dθ
2 S[θ=0:180°] (R³/3)(π/2 + π/2) sinθ dθ
2πR³/3 [-cos(180°)+cos(0°)]
2πR³/3 (1 + 1)
4πR³/3

Replace R with 1.5" and multiply by Yellow, and you get...
4π(1.5)³/3 = 14.137 in³


//notation: since there is no good way to type integral notation, I am using this fairly straightforward format:
S[x=a:b] x dx
which is read "the integral of x dx as x ranges from a to b".

Answer 2

Your radius, height and circumference are not consistent.

Let

Radius = r
Diameter = d
Circumference = c

Then
d = 2r

So the height of the ball is the diameter and would be twice the radius.

If d = 3, then r = 1.5

c = πd = 2πr

If r = 5, then c = 2π5 = 10π = 31.4

Answer 3

If the ball is a sphere, the volume of a sphere is

V = 4/3 (pi) r cubed

Answer 4

One thing that could probably help with this is one Centimeter cubed equals one Milliliter.