Question

What operation does an integral perform?

Answer 1

if any

Answer 2

Integration can be thought of as taking a function space to real values, where the real values can be thought of as "area" (in 2 dimensions) under the curve the function represents.

Answer 3

The function space? I'm sure once I understand these terms I'll get the logic lol

Answer 4

If you think of integration as a function or map, it is simply taking functions to values. Do you understand what I mean?

Answer 5

\(\mathscr{F} = \{f : [0, 1] \to \mathbb{R} \ |\ f \text{ is continuous a.e. on [0, 1]}\}\)

We can now define an operation or function \(\texttt{Int} : \mathscr{F} \to \mathbb{R}\) follows.

\(\texttt{Int}(f) = \int_0^1 f(x) \ dx\)

Answer 6

Here \(\mathscr{F}\) is the space of almost everywhere continuous functions (also known as Riemann integrable functions) and the integral is simply taking them to real values.

Answer 7

Err, the space of almost everywhere continuous functions on [0, 1] (I chose this as an example)

Answer 8

This is probably just too far ahead for me, because I'm just not getting it. :( I saw them and was curious, because I'm trying to learn calculus.

Answer 9

Oh I see, you are just trying to understand Riemann integration? Think of it as area under a curve.

Answer 10

Okay
That makes sense

Answer 11

The Riemann integral starts by partitioning the domain (slicing it) and then summing the areas rectangles (whose height depends on the value of the function somewhere on the interval where the rectangle resides). The Riemann integral is the limit as the partition gets arbitrarily fine.

Answer 12

It is better just to read a text or the Wiki article on it.

Answer 13

http://en.wikipedia.org/wiki/Riemann_integral

Answer 14

Alright. Thanks!