# A little confused about integrals

Tags:
1. Jan 8, 2015

### taylor__hasty

I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function?

If an integral is the area under a curve, why do we even have to find the anti derivative at all?

2. Jan 8, 2015

### PeroK

An integral is the area under the curve of the (original) function.

This area is, however, closely related to the anti-derivative of the function. This is the Fundamental Theorem of Calculus.

3. Jan 8, 2015

### Ray Vickson

The antiderivative (if available) is useful because if $A(a,z)$ is the area under the curve $y = f(x)$ from $x=a$ to $x=z$ (with $a$ some fixed number), and if $F(x)$ is the antiderivative of $f(x)$, then we have:
$$\text{area} = A(a,z) = F(z) - F(a)$$
(Fundamental Theorem of Calculus).

So, if you know the antiderivative function you can use it to calculate the area. That is a lot easier than splitting up the interval $[a,z]$ into 100 billion little intervals and adding up the 100 billion little rectangular areas (which would give a good approximation to the actual area---not necessarily the exact area).

4. Jan 19, 2015

### Svein

Integration can be used to find the area under a curve, but that is just one trivial function of the integration tool.

As to anti-derivatives: The definition is that F is the anti-derivative of f if dF/dx = f. This has one important consequence:

F is differentiable. f does not have to be. I fact, f does not even have to be continuous.