Volume of Football

[HeyHow!]

We have been given the task of finding the volume of a football (elliptical).

i know the area for an ellipse is \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where a=distance from center to major axis x-direction (Half the length of the ball) (a=14cm)

and

b=distance to minor axis y-direction (circumference/2pi) \frac{73}{2\pi}

from there i get confused. i found on a website that
Now if we integrate Area*dx where Area= pi*y^2 (area of a cross
section of the football as given for our ellipse above) between 0 and
5.5 we will obtain half the volume for our ideal football.
This integration results in:
(pi*b^2/a^2)* ((a^3) - ((a^3)/3))
One can simplify this equation into:
(2/3)*pi*a*(b^2)
Remember, this is half the volume of our ideal football. To be more
correct, one would integrate between -5.5 and 5.5. The calculations
work out easier using 0 to 5.5.
i dont understand the Area*dx and where Area= pi*y^2 comes from. Does that mean that pi*y^2 has to be integrated? I also have no idea of how the integration would look like. i do not know what to integrate to get to (pi*b^2/a^2)* ((a^3) - ((a^3)/3)).

any help would be appreciated greatly. i am confused

[Njorl]

Looking at the football from the skinny end you see a circle.

You calculate the volume by slicing the football into a bunch of circular slices and finding their volumes. The volume of each slice is the area of the circle, times its thickness. The area of a circle is pi*r2. The thickness is dx. Each slice occurs at a different value of x. The radius of each circle is the y value at that x determined by the equation of the elipse. (actually, it is the seperation of the |y| from the x-axis of the ellipse, but since your ellipse is on the coordinate system x-axis, the y value is that seperation.)

So, your total volume is the sum of each individual volume which is pi*y2dx.

You re-arrange your ellipse equation to get y2 in terms of x and integrate.

Njorl