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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
Whats the slope of
f(x) = x at
x = 5? Well
1 of course. It’s
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
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
k(x)=5x can be evaluated at
x=7 by drawing a triangle of height
35 and width
7 to get an area of
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
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.