murshid_islam
Nov7-11, 01:37 PM
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 y = x^n, 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):
Figure 19 shows a portion of the curve y = x^n between the points x = 0 and x = a on the x-axis. We imagine that the interval from x = 0 to x = a is divided into an infinite number of subintervals by the points ... K, L, M, N, where ON = a. Then, starting at N and working backward, if these intervals are to form a decreasing geometric progression, we have ON = a, OM = ar, OL = ar2, and so on, where r is less than 1. The heights (ordinates) to the curve at these points are then a^n, (ar)^n, (ar^{2})^n, .... From this it is easy to find the area of each rectangle and then sum up the areas, using the summation formula for an infinite geometric series. The result is the formula,
A_{r} = \frac{a^{n+1}(1-r)}{1 - r^{n+1}}
where the subscript r under the A indicates that this area still depends on our choice of r.
Now I can't get to the final formula. The areas of each rectangle I found are a^{n+1}, (ar)^{n+1}, (ar^{2})^{n+1}, and so on. Their sum,
A_{r} = a^{n+1} + (ar)^{n+1} + (ar^{2})^{n+1} + \cdots
= a^{n+1}\left(1 + r^{n+1} + r^{2(n+1)} + \cdots \right)
= \frac{a^{n+1}}{1 - r^{n+1}}
Where am I getting wrong?
.
Figure 19 shows a portion of the curve y = x^n between the points x = 0 and x = a on the x-axis. We imagine that the interval from x = 0 to x = a is divided into an infinite number of subintervals by the points ... K, L, M, N, where ON = a. Then, starting at N and working backward, if these intervals are to form a decreasing geometric progression, we have ON = a, OM = ar, OL = ar2, and so on, where r is less than 1. The heights (ordinates) to the curve at these points are then a^n, (ar)^n, (ar^{2})^n, .... From this it is easy to find the area of each rectangle and then sum up the areas, using the summation formula for an infinite geometric series. The result is the formula,
A_{r} = \frac{a^{n+1}(1-r)}{1 - r^{n+1}}
where the subscript r under the A indicates that this area still depends on our choice of r.
Now I can't get to the final formula. The areas of each rectangle I found are a^{n+1}, (ar)^{n+1}, (ar^{2})^{n+1}, and so on. Their sum,
A_{r} = a^{n+1} + (ar)^{n+1} + (ar^{2})^{n+1} + \cdots
= a^{n+1}\left(1 + r^{n+1} + r^{2(n+1)} + \cdots \right)
= \frac{a^{n+1}}{1 - r^{n+1}}
Where am I getting wrong?
.