>[!info] Definition >$\huge \let f(x)$ be [[Continuous]] on $\huge [a,b]$. $\huge \let F(x)$ be any [[Anti-Derivative]] of $\huge f(x)$. *Then* $\huge \begin{align} \int_{a}^b f(x)\d{x} &= F(b) - F(a) \\ &=F(x) \vert_{a}^b \end{align}$ The [[Fundamental Theorem of Calculus]] describes the nature of [[Derivative|Derivation]] and [[Integration]], and how they are Opposites. >[!example] >$\huge \begin{align} &\int _{ 0}^1 x^2\d{x} \\ =& \frac{x^3}{3} \bigg\rvert^1_{0} \\ =& \frac{1^3}{3} - \frac{0^3}{3}\\ =& \color{limegreen}\boxed{\frac{1}{3}} \end{align} $