3 ms·

Around "When you integrate a function the result is more continuous and more smooth."

ColinWright 11y ago
So suppose you have a function. Don't draw a nice one, draw one that's jaggy, or even discontinuous. Now start from somewhere and move rightwards, drawing the function of how much area is under the original. You'll find that (except for truly pathological cases) the function you draw changes continuously, never jumping radically, because as you go a little further right you can only get a little more area. So the integral is, in a very real sense, "smoother" than the original. You can turn this around and think of it the other way. If you have the function f(x)=abs(x) and you differentiate it, the derivative has a jump discontinuity at x=0. So differentiating can make things less smooth, and hence integrating makes things more smooth. As it says immediately after the sentence you quote: > When you integrate a function the result is more > continuous and more smooth. In order to get > something out that’s discontinuous at a given > point, the function you put in needs to be > infinitely nasty at that point (technically, > it has to be so nasty it’s not even a function). Does that help?
- grayclhn 11y ago
  Just to add to that (for GP, obviously): if "drawing" seems too abstract or calculating the area on the fly seems too hard, you can get similar intuition by generating a wiggly/crinkly sequence (technical term) of numbers on your computer, then repeatedly take cumulative sums. This can even be done in excel (mash the number keys along a column for the initial function), and then you can graph the resulting sequences as functions. The cumulative sum is analogous to the integral, and the sequence will visibly get smoother. For the derivatives, take the difference repeatedly.
  - daveloyall 11y ago
    Thank you. And to Colin, too.

So suppose you have a function. Don't draw a nice one, draw one that's jaggy, or even discontinuous. Now start from somewhere and move rightwards, drawing the function of how much area is under the original. You'll find that (except for truly pathological cases) the function you draw changes continuously, never jumping radically, because as you go a little further right you can only get a little more area. So the integral is, in a very real sense, "smoother" than the original. You can turn this around and think of it the other way. If you have the function f(x)=abs(x) and you differentiate it, the derivative has a jump discontinuity at x=0. So differentiating can make things less smooth, and hence integrating makes things more smooth. As it says immediately after the sentence you quote: > When you integrate a function the result is more > continuous and more smooth. In order to get > something out that’s discontinuous at a given > point, the function you put in needs to be > infinitely nasty at that point (technically, > it has to be so nasty it’s not even a function). Does that help?

grayclhn 11y ago
Just to add to that (for GP, obviously): if "drawing" seems too abstract or calculating the area on the fly seems too hard, you can get similar intuition by generating a wiggly/crinkly sequence (technical term) of numbers on your computer, then repeatedly take cumulative sums. This can even be done in excel (mash the number keys along a column for the initial function), and then you can graph the resulting sequences as functions. The cumulative sum is analogous to the integral, and the sequence will visibly get smoother. For the derivatives, take the difference repeatedly.
- daveloyall 11y ago
  Thank you. And to Colin, too.