How did Cavalieri get his formula for the area underneath a parabola?

[erocored]

I know he had this ratio:

But how did he get this:

?

 

[erocored]

 

[Mark44]

I know he had this ratio:
View attachment 272042
But how did he get this:
View attachment 272043 ?

The result uses a formula for the sum of squares.
$$\sum_{k=1}^m k^2 = \frac{m(m + 1)(2m + 1)}6$$
Replace $1^2 + 2^2 + \dots + m^2$ in your first equation by the above, and you will get the result shown in the second equation.