Fundamental theorem of calculus

The Fundamental theorem of calculus is a theorem at the core of calculus, linking the concept of the derivative with that of the integral. It is split into two parts.

The first fundamental theorem of calculus states that given the continuous function $f(x)$ , if

$F(x)=\int\limits_a^x f(t)dt$

Then

$F'(x) = f(x)$

The second fundamental theorem of calculus states that:

$\int\limits_a^b f(x)dx=F(b)-F(a)$

The fundamental theorem of calculus has great bearing on practically calculating definite integrals (by taking the antiderivative), unlike other formal definitions such as the Riemann sum.

Proof of the first theorem

For a given $f(t)$ , define the function

$F(x)=\int\limits_a^x f(t)dt$

For any two numbers $x_0,x_0+h\in[a,b]$ , we have

$\begin{align} &F(x_0)=\int\limits_a^{x_0}f(t)dt\\&F(x_0+h)=\int\limits_a^{x_0+h}f(t)dt \end{align}$

It can be shown that

$\int\limits_a^{x_0+h}f(t)dt=\int\limits_a^{x_0}f(t)dt+\int\limits_{x_0}^{x_0+h}f(t)dt$

(The sum of the areas of two adjacent regions is equal to the area of both regions combined.)

Manipulating this equation gives

$\begin{align} F(x_0+h)-F(x_0)&=\int\limits_a^{x_0+h}f(t)dt-\int\limits_a^{x_0}f(t)dt\\ &=\int\limits_{x_0}^{x_0+h}f(t)dt\end{align}$

According to the mean value theorem for integration, there exists an $x_h\in[x_0,x_0+h]$ such that

$F(x_0+h)-F(x_0)=\int\limits_{x_0}^{x_0+h}f(t)dt=f(x_h)\cdot h$

Dividing both sides by $h$ gives

$\frac{F(x_0+h)-F(x_0)}{h}=f(x_h)$

The expression on the left side of the equation is Newton's difference quotient for $F$ at $x_0$ .

Take the limit on both sides of the equation.

$\lim_{h\to0}\frac{F(x_0+h)-F(x_0)}{h}=\lim_{h\to0}f(x_h)$

The expression on the left side of the equation is the definition of the derivative of $F$ at $x_0$ .

$F'(x_0)=\lim_{h\to 0}f(x_h)$

To find the other limit, we use the squeeze theorem.

$\begin{align}&x_h\in[x_0,x_0+h]\quad\Rightarrow\quad x_0\le x_h\le x_0+h\\ &\lim_{h\to0}x_0=\lim_{h\to0}\Big[x_0+h\Big]=x_0\end{align}$

Therefore, according to the squeeze theorem,

$\lim_{h\to0}x_h=x_0$

We get

$F'(x_0)=\lim_{x_h\to x_0}f(x_h)$

The function $f$ is continuous at $x_h$ , so the limit can be taken inside the function. Therefore, we get

$F'(x_0)=f(x_0)$

which completes the proof. $\blacksquare$