Hi,

I've just started working through volume one of Tom M Apostol's Calculus.

The first chapter gives a brief history of the method of exhaustion and attempts to show how this is used to show that the area of a parabolic segment is exactly 1/3 of the area of the rectangle enclosing it.

However when it comes to explaining equations 1.1 and 1.2, I dont understand how (n-1)^2 comes into equation 1.2.

I have attatched the parabolic segment and explanation to the problem. I know this will be some fundamental gap in my understanding of basic math but i cant follow through and understand how equations 1.1 and 1.2 differ.

 

Thank you in advance for any help.

parabolic segment

Explanation

Dr. Finlay, the key difference is that for 1.1, the 'above' rectangles, the height (or altitude as they call it) is measured at the right hand side of the rectangle, where as for the 'below' rectangles we measure from the left. This means that the very first below rectangle has a height measured at zero. Hence the height of the first 'below' rectangle is 0 and dose not count to the total, with this rectangle missing we are left with n-1 rectangles that contribute to the total.

AH! I understand that now. Thanks alot for your help. I knew it would be something simple that i missed

Always a pleasure