fundamental_theorem

Fundamental Theorem of Calculus

Statement

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations. It has two parts:

• First Theorem: If $F(x)={\int }_a^{x}f(s)ds$, then $F^{\prime }(x)=f(x)$.
• Second Theorem: For any antiderivative $F$ of $f$, ${\int }_a^{b}f(s)ds=F(b)-F(a)$.

Assumptions

• $f$ is continuous on the closed interval $[a,b]$.
• $F$ is any function satisfying $F^{\prime }(x)=f(x)$ on $[a,b]$ (an antiderivative of $f$).

Proof Sketch

First Theorem. The integral ${\int }_a^{x}f(s)ds$ accumulates area as $x$ increases. Incrementing $x$ by a small $h$ adds a thin strip of width $h$ and height $\approx f(x)$; dividing by $h$ and taking the limit recovers $f(x)$.

Second Theorem. Because $F(x)={\int }_a^{x}f(s)ds+C$ is the general antiderivative, evaluating at the endpoints and subtracting eliminates $C$, yielding ${\int }_a^{b}f(s)ds=F(b)-F(a)$.

Full Proof

First Fundamental Theorem. Define the accumulation function

$$y(x)={\int }_a^{x}f(s)ds.$$

By the definition of the derivative,

$$y^{\prime }(x)=\underset{h\to 0}{\lim }\frac{{\int }_a^{x+h}f(s)ds-{\int }_a^{x}f(s)ds}{h}=\underset{h\to 0}{\lim }\frac{{\int }_x^{x+h}f(s)ds}{h}.$$

Since $f$ is continuous, for small $h$ the integral ${\int }_x^{x+h}f(s)ds$ approximates a rectangle of width $h$ and height $f(x)$. Therefore

$$\frac{d}{dx}{\int }_a^{x}f(s)ds=f(x).$$

Second Fundamental Theorem. By the First Theorem, $y(x)={\int }_a^{x}f(s)ds$ is an antiderivative of $f$, so the general antiderivative is

$$F(x)={\int }_a^{x}f(s)ds+C.$$

Evaluating at $x=a$ gives $F(a)=0+C$, so $C=F(a)$. Evaluating at $x=b$:

$$F(b)={\int }_a^{b}f(s)ds+F(a),$$

and therefore

$${\int }_a^{b}f(s)ds=F(b)-F(a).$$

Equivalently,

$${\int }_a^b{f}^{\prime }(x)dx=f(x){|}_a^b=f(b)-f(a).$$

Notes / Intuition

• Differentiation and integration are inverse operations; the theorem is the formal bridge between them.
• Every continuous function has an antiderivative (given by the accumulation function), even when no closed-form expression exists.
• The Second Theorem justifies the standard technique of computing definite integrals by finding an antiderivative and evaluating at the endpoints, rather than computing a limit of Riemann sums directly.
• Geometrically, the First Theorem says the rate of change of accumulated area equals the height of the curve at the moving boundary.

Links

• Integrals