# Definition of an integral

1. Jul 19, 2010

### delve

Could someone tell me what an integral actually does? I understand that it has something to do with adding infinitely many things, but as you can see, this understanding is vague at best. Please help me. Thanks.

2. Jul 19, 2010

### Char. Limit

Integration is a method of calculating infinitely many slivers of infinitely tiny areas between a function and the x-axis, then summing up these slivers. Indefinite integration has been proven to be the opposite of differentiation.

3. Jul 19, 2010

### delve

Ok, this makes sense, thank you! :)

4. Jul 22, 2010

### paulfr

Integration is just multiplication when one of the inputs is changing.

Remember that Calculus is the Mathematics of Change.

Integral of 3 dx over the interval from 0 to 4 is = 3 x [ 4 - 0 ] = 12

Now suppose we want to do this multiplication for y = f (x) = 2x + 5 rather than y = f(x) = 3
We have to divide up the area
[recall the 2 dimensional interpretation of integration as the area under a curve]
in to small slices and add up all the areas.
This is consistent with the fact that multiplication is nothing more than repeated addition.
3 x 4 = 12
3 x 4 = 3 + 3 + 3 + 3
3 x 4 = 4 + 4 + 4

The smaller/narrower the slices [ dx ] the more accurate the result.
In the limit as delta x goes to zero we have the integral which is the product of f(x) and dx over the interval.

Hope that helps