THE FUNDAMENTAL THEOREM OF CALCULUS The Fundamental Theorem

THE FUNDAMENTAL THEOREM OF CALCULUS The Fundamental Theorem of Calculus is appropriately named because it establishes connection between the two branches of calculus: differential calculus and integral calculus. DEFINITION A function Example is called an antiderivative of if Let: Find the an antiderivative

THE FUNDAMENTAL THEOREM OF CALCULUS, PART 2 Example: Evaluate the integral Find the area under the curve from x-0 to x=1

THE FUNDAMENTAL THEOREM OF CALCULUS, PART 2 Example: Evaluate the integral Find the area under the curve from x-0 to

THE FUNDAMENTAL THEOREM OF CALCULUS, PART 2 Example: Evaluate the integral

THE FUNDAMENTAL THEOREM OF CALCULUS Define: Example:

THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1 Example: Note: Find the derivative of the function Using Leibniz notation

THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1 Note: Using Leibniz notation Example: Find

THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1 Note: Using Leibniz notation Example: Note: Find

THE FUNDAMENTAL THEOREM OF CALCULUS Note:

THE FUNDAMENTAL THEOREM OF CALCULUS Note which says that if f is integrated and then the result is differentiated, we arrive back at the original function Note This version says that if we take a function , first differentiate it, and then integrate the result, we arrive back at the original function

Total Area Example Evaluate:

Total Area

Total Area Example Find the area of the region between the x-axis and the graph of What is the difference between these two examples

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