A little confused about integrals

[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?

 

[PeroK]

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?

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.

 

[Ray Vickson]

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?

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).

 

[Svein]

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?

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.