In calculus, we first learn the derivative operator. It takes a smooth function and an x
-value and it returns the slope of the function at that x
-value.
Whats the slope of f(x) = x
at x = 5
? Well 1
of course. It’s 1
everywhere!
How about g(x) = ln x
at x = 17
? If you draw it really accurately you see that it’s 1/17
. (This has to do with ln x
and 1/x
kind of being flips of eachother)
The integral takes a not-necessarily-smooth function and an x
-value, and returns the area between the function and the x-axis between 0 and that point.
How much area is under h(x) = Sqrt(3-x^2)
from 0
to 3
? That’s a quarter of a circle of radius 3
, so the area is pi*r^2 / 4 = pi*9/4
which is about 7
.
More simply k(x)=5x
can be evaluated at x=7
by drawing a triangle of height 35
and width 7
to get an area of 122.5
.
The fundamental theorem of calculus states that these two operators are inverse functions; the derivative of the integral of f(x)
is always that same f(x)
. What?
To understand this, consider at what rate the area is changing. If the function is tall at some x
, then the area grows quickly there. If the function is short, then every inch forwards has nearly no gains to the surface.
Another way of stating this: The rate of change of the area is the function.
Kalid Azad’s more thorough article on the same topic
Public Domain Dedication.