The first fundamental theorem of calculus states that given the continuous function , if
The second fundamental theorem of calculus states that:
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,
The function is continuous at , so the limit can be taken inside the function. Therefore, we get
which completes the proof.