How integration is continuous summation? - Answers

Well look at it this way. To find the exact area under a curve between two points (call them a and b), one can find the definite integral between the points. But we could also find it approximately.Let a=(x1, y1) and b=(x2, y2). Let f be the function that the curve represents. So the point a=(x1,y1) on the curve represents the same thing as f(x1)=y1, that is applying function f to x1 to get result y1.Now we can also say that at x1 the curve has height y1. This is good for our area problem because we know how that for a rectangle, area = height multiplied by width. A width will just be a difference in x value.The horizontal distance between a and b is a width, = x2-x1. Can we just multiply the height by this width? Not really, because for any curve that isn't a straight horizontal line, the height will vary along that width, and so it isn't a rectangle.But what if we divided up the area under the curve into thin vertical strips with equal width? Then these strips would almost be rectangles, so we could find the area of all of them individually and then add them to get a good approximation to the area. This is where the summation comes in.Think of it another way. You have a curve graphed on a computer monitor. How would you calculate the area underneath it in pixels (dots on the computer screen)? Perhaps the most obvious way is to start at the first pixel of the curve and count how many pixels are underneath it, then do that for each pixel of the curve, adding up all the counts.If we assume the curve is smooth, then the thinner we make the strips, the better the approximation is. Smooth means that if the more we zoom on a section of the curve, the more it appears to be a straight line. If it does appear that way then we can get the exact area by a triangle on top a rectangle. But we can ignore the area of the triangle because it will be vanishingly small because we have zoomed in so much.In fact if we take it to the limit of an arbitrarily thin strip, we will get the integral and an exact answer to the area under the curve problem. It is said to be a continuous summation because it is summing the area in the way just described, and is continuous in that it is smooth, not chunky and blocky like with strips of definite thickness.This concept of a limit of arbitrary smallness is the hardest concept to grasp in the calculus, but once you get it, you can understand all of the calculus with an ease you wouldn't have thought possible at first. Then it is just a matter of practice and memorizing to get good at it.