Mathematics

The Fundamental Theorem of Calculus connects derivatives back to integrals: if $F'(x) = f(x)$, then $F$ is an antiderivative of $f$, and you can recover it by integrating. The key is using the initial condition $F(a) = y_0$ as an anchor point—it tells you the constant of integration. Think of the definite integral from $a$ to $x$ as measuring the accumulated change in $F$, then add back the starting height $y_0$ to get the full picture.

The Fundamental Theorem of Calculus connects derivatives back to integrals: if $F'(x) = f(x)$, then $F$ is an antiderivative of $f$, and you can recover it by integrating. The key is using the initial condition $F(a) = y_0$ as an anchor point—it tells you the constant of integration. Think of the definite integral from $a$ to $x$ as measuring the accumulated change in $F$, then add back the starting height $y_0$ to get the full picture.