MTH1170 The Fundamental Theorem of Calculus

The FTC The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral. The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the indefinite integral of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem guarantees the existence of antiderivatives for continuous functions. The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its antiderivatives.

Part 1 Examining the following graph of the function f(t) we can see that it is continuous on the interval [a, b]:

Part 1 Using the concept of the Definite Integral we can write the following:

Part 1 If we differentiate both sides of this equation we get:

Part 1 From this we can realize two things: First: That every continuous function has an antiderivative.  Second: There is a connection between differentiation, integration, and antiderivatives.

Part 2

Part 2 This provides a connection between antiderivatives and definite integrals. This is how we calculate definite integrals.

Properties of the Definite Integral

Example:

Example

Example