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.