is there a difference between integrating a function between two limits and summing a function and if so where does the difference come from and when would you use each method in real life situations

  • $\begingroup$ What do you mean by "summing a function"? $\endgroup$ – fosho Jan 16 '16 at 16:44
  • $\begingroup$ the sigma notation $\endgroup$ – AlexanderRD Jan 16 '16 at 16:45
  • $\begingroup$ Related: math.stackexchange.com/questions/184979/… and math.stackexchange.com/questions/247212/… $\endgroup$ – user174622 Jan 16 '16 at 16:47
  • $\begingroup$ do you have any background in measure theory? If yes, have a look at $\ell^p$ and $L^p$ spaces. And for an application it's very interesting to have a look at the expectation value of discrete and continuous random variables. $\endgroup$ – noctusraid Jan 16 '16 at 16:53
  • $\begingroup$ well "summing a function" means to actually sum the values of the function at each point. integrating is summing the infinitesimal areas at each x between the limits. $\endgroup$ – Airdish Jan 16 '16 at 16:58

There are multiple differences, but the main thing to note is that summation involves discrete values whereas integration involves continuous values.

Something else to note is that integration is just summation over infinite values.

As for in "real life" situations, if your professor told you to count the number of students in your lecture hall, you would count and report to your professor. However, if your professor told you to count how many water particles are in the ocean, then realistically, you would estimate the number of water particles in a smaller area and integrate all of those smaller areas. This will give you a close approximation of the total number of water particles.

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