# Applying the "Cavalieri Principle" to approximate integration

(Apologies for asking another question based on Julian Havil's "The Irrationals")

On page 86 of Havil's "The Irrationals" the author outlines how John Wallis approximated integration by applying a variant of the "Cavalieri Principle" (that two solids are of equal volume if the areas of their cross sections are everywhere equal) as applied to two dimensions.

He outlines this in this fashion - for a "positive value continuous function $$f(x)$$ defined on the positive real axis with the maximum value of $$M_N$$ on the interval $$[0,N]$$":

$$\int_0^1 f(x)\,dx = \lim_{N\rightarrow \infty} \frac{\sum^N_{r=0} f(r)}{\sum^N_{r=0}M_N} = \lim_{N \rightarrow \infty}\frac{\sum^N_{r=0} f(r)}{M_N(N+1)}$$

And writes "In words, add up the lengths of the vertical strips from the $$x$$-axis to the curve and then divide this by the same order approximation of the area of the enclosing rectangle."

What has totally flummoxed me is the idea that the area under the curve of a function between 0 and 1 is related to its value at infinity. I can see that we can approximate the value of the integral by summing the values of the function and then dividing by the interval - but not the sum presented here: could someone explain the reasoning here, please?

• No problem, your discomfort is reasonable, the formula is just wrong. Could be fixed, by a scaling/descaling procedure, but the details are different. Commented Aug 5, 2013 at 17:37
• When I read that my first reaction was that I must have copied it down wrong - but I have rechecked and that is how it is printed in the book :( Commented Aug 5, 2013 at 21:48
• I managed to check also, the Google preview carried it. Commented Aug 5, 2013 at 21:53
• I have spotted a few misprints in the book, but the suggestion here is of an outright error in methodology? Commented Aug 5, 2013 at 21:56
• It can be fixed, we want a Riemann-type sum. Commented Aug 5, 2013 at 22:00

This page - http://www.math.wpi.edu/IQP/BVCalcHist/calc1.html - gives an excellent explanation of Cavalieri's and Wallis's thinking.