I am currently reading the book "e: The Story of a Number" by Eli Maor. And I got stuck at something. In chapter 7 of the book, the author described the method Fermat used to calculate areas under curves of the form [itex]y = x^n[/itex], where n is a positive integer. I am quoting the relevant bit here (sorry, I can't show the figure, but from the description, you can easily receate it):

Now I can't get to the final formula. The areas of each rectangle I found are [itex]a^{n+1}, (ar)^{n+1}, (ar^{2})^{n+1},[/itex] and so on. Their sum,

You are using the ordinates to get the heights of the rectangles. I think the formula is based on using the averages of the adjacent ordinates to get the rectangle heights,