GoKawiilApril 22, 2026
Fundamental Theorem of Calculus
Although the notion of area is intuitive, its mathematical treatment requires a rigorous definition. This post introduces the Riemann integral, and proves the fundamental theorem of calculus—a beautiful result that connects integrals and derivatives.
Riemann integral §
Given a bounded1 function \(f:[a,b]\to\mathbb{R}\), we can approximate the area under its graph by rectangles. Choose a partition of its domain
\[ \mathcal{P}=\{x_0,x_1,\ldots,x_n\mid a=x_0<x_1<\cdots<x_n=b\}. \]
For each subinterval \([x_{k-1},x_k]\), define the width \(\Delta x_k=x_k-x_{k-1}\), and let \(m_k\) and \(M_k\) denote the infimum and supremum of \(f\) on that subinterval. The lower and upper sums are
\[ L(f,\mathcal{P})=\sum_{k=1}^{n}m_k\Delta x_k, \qquad U(f,\mathcal{P})=\sum_{k=1}^{n}M_k\Delta x_k. \]
We define \(f\) to be Riemann integrable2 on \([a,b]\) iff for every \(\varepsilon>0\) there exists a partition \(\mathcal{P}\) such that \(U(f,\mathcal{P})-L(f,\mathcal{P})<\varepsilon\), in which case
\[ \int_a^b f =\sup_{\mathcal{P}}L(f,\mathcal{P}) =\inf_{\mathcal{P}}U(f,\mathcal{P}). \]
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