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 , if

Then

The **second fundamental theorem of calculus** states that:

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 , define the function

For any two numbers , we have

It can be shown that

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

Manipulating this equation gives

According to the mean value theorem for integration, there exists an such that

Dividing both sides by gives

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

Take the limit on both sides of the equation.

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

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

Therefore, according to the squeeze theorem,

We get

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

which completes the proof.