Example 2.11. Integrating a step function.
Consider the step function
\begin{equation*}
f(t)=
\begin{cases}
2, & 0 \le t \lt 1,\\[4pt]
-1, & 1 \le t \lt 3,\\[4pt]
1, & 3 \le t \le 4.
\end{cases}
\end{equation*}
Section 2.3 Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) forges the bridge between differentiation and integration. Given a function \(f(x)\) on a closed interval \(I\) and a point \(a \in I\text{.}\) One can consider the function \(F_a(x) = \int_a^x f(t) dt\text{,}\) i.e. the "signed area" from \(a\) to \(x\) underneath the graph \(y=f(x)\text{.}\) Even though \(F_a(x)\) may not exist for an arbitrary \(f(x)\text{,}\) it does exist, for example, when \(f(x)\) is piecewise continuous. A version of the FTC asserts that \(F' = f\) at the continuous points of \(f\text{.}\) In particular, if \(f\) is continuous on \(I\text{,}\) then the \(F(x)\) thus defined is an anti-derivative of \(f(x)\) (the one that vanishes at \(a\)). In addition, if \(F\) is any antiderivative of a continuous \(f\) on \(I\text{,}\) then the definite integral is evaluated by \(\int_a^b f(x)\,dx = F(b)-F(a)\text{.}\) Together, these statements tell us that “differentiation undoes integration”, and they provide a practical way to compute integrals.
