Calc05_3.ppt - 5.3 Definite Integrals and Antiderivatives Organ Pipe Cactus National Monument Arizona Photo by Vickie Kelly 2009 Greg Kelly Hanford High

# Calc05_3.ppt - 5.3 Definite Integrals and Antiderivatives...

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5.3 Definite Integrals and Antiderivatives Organ Pipe Cactus National Monument, Arizona Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2009
Your textbook gives rules for working with integrals, the most important of which are: 2. 0 a a f x dx If the upper and lower limits are equal, then the integral is zero. 1. b a a b f x dx f x dx  Reversing the limits changes the sign. b b a a k f x dx k f x dx 3. Constant multiples can be moved outside.
1. 0 a a f x dx If the upper and lower limits are equal, then the integral is zero. 2. b a a b f x dx f x dx  Reversing the limits changes the sign. b b a a k f x dx k f x dx 3. Constant multiples can be moved outside. b b b a a a f x g x dx f x dx g x dx 4. Integrals can be added and subtracted.
b b b a a a f x g x dx f x dx g x dx 4. Integrals can be added and subtracted. 5. b c c a b a f x dx f x dx f x dx Intervals can be added (or subtracted.) a b c y f x
The average value of a function is the value that would give the same area if the function was a constant: 2 1 2 y x 3 2 1 2 A x dx 3 3 1 6 x 27 6 9 2 4.5 4.5 Average Value 1.5 3 Area 1 Average Value Width b a f x dx b a 1.5
The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal to the average value. Mean Value Theorem (for definite integrals) If f is continuous on then at some point c in , , a b , a b 1 b a f c f x dx b a Note: When a problem on the AP Exam refers to the “Mean Value Theorem” without specifying which one, they mean the Mean Value Theorem for Derivatives .