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Ch. 6 – The Definite Integral
6.3 – Definite Integrals and Antiderivatives
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Properties of Integrals
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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 = together. Does this answer make sense?
![Ex: Find the average value of f(x) = 2 + xsinx on [0, 7π/6].](http://slideplayer.com/slide/13419025/81/images/3/Ex%3A+Find+the+average+value+of+f%28x%29+%3D+2+%2B+xsinx+on+%5B0%2C+7%CF%80%2F6%5D..jpg "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 = together. Does this answer make sense")
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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…
![Mean Value Theorem for Integrals: If f is continuous on [a, b], then at some point c in [a, b],](http://slideplayer.com/slide/13419025/81/images/4/Mean+Value+Theorem+for+Integrals%3A+If+f+is+continuous+on+%5Ba%2C+b%5D%2C+then+at+some+point+c+in+%5Ba%2C+b%5D%2C.jpg "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…")
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Ex: Evaluate the following:
Fundamental Thm. Of Calculus, Part II: Ex: Evaluate algebraically: Antiderivative of sinx is –cosx…
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Ex: Evaluate algebraically:
Antiderivative is -1/x… Flip the upper and lower bounds to lose the negative Antiderivative is x6 – 3x4 – 3x3 …
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