fundamental theorem of calculus

the fundamental theorem of calculus

1. A theorem of calculus consisting of two parts:^[1]
   1. For a continuous function ${\displaystyle f}$, an antiderivative or indefinite integral of ${\displaystyle F}$ can be obtained as the integral of ${\displaystyle f}$ over an interval with a variable upper bound, or: ${\displaystyle F(x)=\int _{a}^{x}f(t)\,dt}$.
   2. The definite integral of a continuous function ${\displaystyle f}$ over a closed interval ${\displaystyle [a,\,b]}$ is equal to the net change of any antiderivative ${\displaystyle F}$ over ${\displaystyle [a,\,b]}$, or: ${\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a)}$.

• divergence theorem
• Green's theorem
• Stokes' theorem