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

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 Theorem

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.