Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects two central ideas in calculus: accumulation and rate of change. It tells us how integrals and derivatives are two sides of the same coin. 🔁

What the Theorem Says

Let’s say you have a function f that is continuous (no jumps or breaks) on the interval [a, b]. Then if we define a new function:

F(x) = \int_a^x f(t)dt

This new function F(x) has as a derivative the original function:

F’(x) = f(x)

Taking the derivative of the area function gives you back the original function f(x).

Also, if F is any antiderivative of f (meaning F’(x) = f(x)), then the definite integral from a to b is:

\int_a^b f(x),dx = F(b) - F(a)

To find the total area (or accumulation) from a to b, just plug into the antiderivative and subtract.