Finding the Area Under a Parabola Using Archimedes' Method

[Sculptured]

I've started with Apostol's first volume of calculus and I'm already having a problem with the first exercise in the book. Weak. It asks to use Archimedes' method to find the area under a parabola from 0<x<b for a parabola where y = a*b^2 + c. Using the summation of upper/lower rectangles of width b/n and height a(k*b/n)^2 +c. Now each rectangle has an area of a(k^2)(b/n)^3 + (bc)/n. k goes from 0 to n-1 for the lower rectangles and from 1 to n for the upper rectangles. My problem comes when summing up all of the rectangles using the arithmetic summation. I get the first portion correct, but end up with (bc)/n instead of bc as the second term in the area formula. Any way that I'm approaching this incorrectly or should approach this?

 

[StatusX]

The sum bc/n+bc/n+...+bc/n (n times) is bc.

 

[Sculptured]

Ah, right. Stupid of me to miss that result.