Difference between $\sum$ and $\int$

What is the difference between summation and integration. I have heard that summation is nothing but definite integration. But $ \sum_{i=1}^{10}i $ and $ \int_{i=1}^{10}i $ yields very different values.

$ \sum_{i=1}^{10}i = 55 $

but

$ \int_{i=1}^{10}i = 49.5 $

So where's the catch. I have seen a lot of questions regarding this, but couldn't find a practical example.

Solution 1:

Since a picture says more than words ...

enter image description here

Solution 2:

In case $f$ is a simple function such that $f|_{[n,n+1)}$ is constant $f(n)$ for all $n\in\Bbb Z$, then we have $$ \int_0^N f = \sum_{0\le n<N} f(n).$$ In your example the function to integrate was the identity $i\mapsto i$, this is not a simple step function as above, and the integration interval should also be of length $10$ instead of $9$ for a better approximation.

If we choose finer resolutions than $1$, say $\delta$, and suppose now that $f$ is a simple function such that $f|_{[n\delta,(n+1)\delta)}$ is constant for all $n\in\Bbb Z$ then what will we have for $ \displaystyle\int_0^{N\delta} f(x)dx = ?$

And so on, finally every integrable function can be approximated by these kind of simple functions, and as $\delta$ tends to $0$, we'll get the exact integral as limit.

Solution 3:

In simple terms, a summation is the calculation of the sum of discrete values. Consider this example:

$\displaystyle\sum\limits_{i=0}^{10} i = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55$

Integration is the sum of an infinite number infinitely small values. So, I could not rewite an integration problem as the sum of some numbers, like this:

$\int_0^{10} \! x\, \mathrm{d} x, = n + n + ... + n$

Integration is the sum of some numbers, but the $n$ values here cannot be listed because they are infinitely small and non-discrete. It's far more of an abstract concept.

Instead, I think of integration as the area under the graph of the function.

enter image description here