The Fundamental Theorem of Riemann Integral Calculus Part 1

# The Fundamental Theorem of Riemann Integral Calculus Part 1

Perhaps one of the most famous "fundamental" theorems in mathematics are the Fundamental Theorems of Calculus that are first introduced in an introductory calculus course. There are two parts of the fundamental theorem.

The first part of the Fundamental Theorem of Calculus states that if $f$ is a continuous function on the interval $[a, b]$ and $F(x) = \int_a^x f(t) \: dt$ then:

(1)
\begin{align} \quad F'(x) = f(x) \end{align}

The second part of the Fundamental Theorem of Calculus states that if $f$ is continuous on the interval $[a, b]$ and $F$ is any antiderivative of $f$ then:

(2)
\begin{align} \quad \int_a^b f(x) \: dx = F(b) - F(a) \end{align}

We will first derive the Fundamental Theorem of Calculus Part 1 which we will call the Fundamental Theorem of Riemann Integral Calculus Part 1. We derive The Fundamental Theorem of Riemann Integral Calculus Part 2 subsequently.

 Theorem 1 (The Fundamental Theorem of Riemann Integral Calculus Part 1) If $f$ is continuous on $[a, b]$ and $F(x) = \int_a^x f(x) \: dx$ then $F'(x) = f(x)$ on $[a, b]$.
• Proof: Let $f$ be any continuous function on $[a, b]$. Let $\alpha (x) = x$. Then $\alpha$ is an increasing function on $[a, b]$ and is clearly of bounded variation on $[a, b]$. Furthermore, $\alpha'(x) = 1$ exists for all $x \in [a, b]$. Clearly $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$, and so add of the hypotheses from part (c) of the theorem on the Riemann-Stieltjes Integral Defined Functions page is satisfied, so for the function:
(3)
\begin{align} \quad F(x) = \int_a^b f(x) \: dx \end{align}
• We have that $F'(x)$ exists for all $x \in [a, b]$ and so for all $x \in [a, b]$ we have that:
(4)
\begin{align} \quad F'(x) = f(x)\alpha'(x) = f(x) \cdot 1 = f(x) \quad \blacksquare \end{align}