Consider the function f(x)=x2 on the interval [0,b]. Let n be a large positive integer equal to the number of rectangles that we will use to approximate the area under the curve f(x)=x2. If we divide the interval [0,b] by n equal subintervals by means of n-1 equally spaced points, then we have point x1=b/n, x2=2b/n, ..., xn-1=(n-1)b/n.
From this, I understand that the base of each rectangle will be b/n. What I'm not clear on is how to determine the heights of each rectangle. My book says that if we use the upper sums (rectangles that reach just above the curve), then the heights are f(x1)=(b/n)2, f(x2)=(2b/n)2, ..., f(xn)=(nb/n)2.
Can someone please explain? I can't consult a professor or other students because I'm studying calculus on my own. Thanks.