4.4 The Fundamental Theorem of Calculus

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Presentation transcript:

4.4 The Fundamental Theorem of Calculus Evaluate a definite integral using the Fundamental theorem of Calculus. Understand and us the Mean Value Theorem for Integrals. Find the average value of a function over a closed interval. Understand and use the Second Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus relates differentiation and integration. Informally, the theorem states that differentiation and integration are inverse operations. Differentiation: Slope of tangent line = Δy/Δx Definite integration: Area under curve = ΔyΔx

∫ f(x)dx = F(b) – F(a) Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then ∫ f(x)dx = F(b) – F(a) Proof http://archives.math.utk.edu/visual.calculus/4/ftc.9/ http://mathforum.org/library/drmath/view/52018.html http://faculty.uml.edu/spennell/92.142/FTC_proof_final.pdf

Guidelines for using Fundamental Thm. Of Calculus If you can find an antiderivative of f, you can now evaluate a definite integral without the limit process!!! Use the following notation It is not necessary to include a constant of integration C in the antiderivative because

A definite integral involving absolute value A definite integral involving absolute value. Calculator Y₁- enter function; graph 2nd calculate- #7 Enter bounds from the integral.

Average value of a Function If f is integrable on the closed interval [a,b], then the average value of f on the interval is