Ch. 6 – The Definite Integral

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Presentation transcript:

Ch. 6 – The Definite Integral 6.3 – Definite Integrals and Antiderivatives

Properties of Integrals

Ex: Find the average value of f(x) = 2 + xsinx on [0, 7π/6]. Average (Mean) Value: If f is integrable on [a, b], then its average value over that interval is Find the integral, then divide by the interval width! Ex: Find the average value of f(x) = 2 + xsinx on [0, 7π/6]. Use the formula to set up the equation, then use your calculator to solve! Graph f and y = 2.730 together. Does this answer make sense?

Mean Value Theorem for Integrals: If f is continuous on [a, b], then at some point c in [a, b], Any continuous function must reach its average value over an interval! Fundamental Theorem of Calculus, Part I: Let F(t) represent the antiderivative of f(t). Then…

Ex: Evaluate the following: Fundamental Thm. Of Calculus, Part II: Ex: Evaluate algebraically: Antiderivative of sinx is –cosx…

Ex: Evaluate algebraically: Antiderivative is -1/x… Flip the upper and lower bounds to lose the negative Antiderivative is x6 – 3x4 – 3x3 …