It doesn't seem right, but I could have sworn I remember seeing instances in statistics where a summation was literally just swapped with an integral sign without anything else changing, except adding a dx. Can you really just swap a sum and an integral? I mean, the sum of a linear variable k is a second degree polynomail, and the integral of a continuous variable k would be a 2nd degree polynomial, so maybe there's something to it.

# Can you simply replace an integral with a summation?

Started by SFNQuestions, Feb 14, 2017

2 replies to this topic

### #1

Posted 14 February 2017 - 01:03 AM

### #2

Posted 14 February 2017 - 06:43 PM

The answer is yes, under certain circumstances.

The conventional way to define the Riemann definite intgral of a function over a close interval is to divide this interval into a number of non-overlapping interval

where .

You form the so-called Riemann sum where denotes a point in the interval .

Now you let the number of intervals increase without bound, so that , then provided the limit of the Riemann sum exists, then this goes over to the integral

The conventional way to define the Riemann definite intgral of a function over a close interval is to divide this interval into a number of non-overlapping interval

where .

You form the so-called Riemann sum where denotes a point in the interval .

Now you let the number of intervals increase without bound, so that , then provided the limit of the Riemann sum exists, then this goes over to the integral

### #3

Posted 17 February 2017 - 12:13 PM

You can **approximate **an integral by a Riemann sum. Is that what you mean by "replace"?

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