area under a straight line the calculus way, the book gets by means of the regular definit integral method as oppossed to the fundamtenal theorem of calculus method, S of n = 2deltaX+2deltaX+2(deltaX)^2+2deltaX+2x2(deltaX)^2+ . . . +2deltaX+2(n-1)(deltaX)^2 from the general terms arrived at prior. The book then notes there are 2deltaX's in each term, and there are 2n deltaX's. But then, it factors out, not 2deltaX's, but 2(deltaX)^2, and that becomes times the progression [1+2 . . . + (n-1)] As indicated, I see where it gets the 2ndeltaX as an expression for all the 2deltaX's, but then, it just quite mysteriously to me drops it, and factors out 2(deltaX)^2's! Which I'd understand if that was the common term, but where and how did the 2ndeltaX term get taken out?