4009 Fundamental Theorem of Calculus (Part 2) BC CALCULUS

The Indefinite Integral (Antiderivative) finds a Family of Functions whose derivative is given. Given an Initial Condition we find the Particular Function

The Definite Integral as a Particular Function: Evaluate at Evaluate the Definite Integral for each of these points. The Definite Integral is actually finding points on the Accumulation graph. Evaluate the definite integral.

Since A(x) is a function, what then is the rate of change of that function? In words, integration and differentiation are inverse operations

2 nd Fundamental Theorem of Calculus Given:, we want to find Note: a is a constant, u is a function of x ; and the order matters! 2 nd Fundamental Theorem of Calculus: If f is continuous on an open interval, I, containing a point, a, then for every x in I :

Demonstration: find In Words:

Example: Find and verify:

Example: Find without Integrating:

THE COMPOSITE FUNCTION If g(x) is given instead of x: = In words: Substitute in g(x) for t and then multiply by the derivative of g(x)…exactly the chain rule (derivative of the outside * derivative of the inside)

THE COMPOSITE FUNCTION If, (a composite function) then In Words:

Demonstration: Find: In Words:

Example : Find without Integrating: If, solve for

Example: Rewriting the Integral Find without integrating: Show middle step

Example: Rewriting the Integral - Two variable limits: Find without Integrating: break into two parts..... chose any number in domain of for a and rewrite into required form.

Last Update: 1/25/11 Worksheet