Mathematics
The Fundamental Theorem of Calculus and Its Consequences
Quick fact
The Fundamental Theorem of Calculus has two parts: one says that the derivative of an integral (with a variable upper limit) gives back the original function; the other says that a definite integral can be evaluated by subtracting the antiderivative at the endpoints. This turns computing areas into finding antiderivatives.
Why this is interesting
You've probably heard that differentiation and integration are opposites—but have you ever wondered why that's true? This surprising link is the Fundamental Theorem of Calculus, and it's the secret behind almost everything calculus can do.
Read the full explanation
Understanding The Fundamental Theorem of Calculus and Its Consequences
Imagine you're tracking the speed of a car. The speedometer tells you how fast you're going at each instant—that's the derivative of your position. If you want to know the total distance traveled over a trip, you could add up small distances over each tiny time interval—that's an integral. The Fundamental Theorem of Calculus says these two ideas are two sides of the same coin: integration is the reverse of differentiation. More precisely, if you start with a function f and define its accumulation function F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x). This means that the rate at which the accumulated area grows equals the height of the function. That's why the area under a curve can be found by taking an antiderivative.
A deeper explanation
The mechanism behind the theorem is based on the local behavior of the accumulation function. Let F(x) = ∫[a to x] f(t) dt. Then the derivative F'(x) = lim[h→0] (F(x+h)−F(x))/h. But F(x+h)−F(x) = ∫[x to x+h] f(t) dt, which is the area of a thin strip. For small h, this area is approximately f(x)·h, so the quotient is approximately f(x). Taking the limit gives F'(x) = f(x). This is Part 1. Part 2 follows: for any antiderivative F of f (i.e., F' = f), we have ∫[a to b] f(x) dx = F(b) − F(a). This is because any two antiderivatives differ by a constant, and the accumulation function is one specific antiderivative. This theorem is what makes integration practical: instead of summing many tiny quantities, you just find an antiderivative. Its consequences extend to solving differential equations, computing areas and volumes, and understanding the relationship between local and global behavior of functions.