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5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington Page 269 gives rules for working with integrals, the most important of which are: 1. 2. 3. b a a b a a f x dx f x dx a b Reversing the limits changes the sign. f x dx 0 If the upper and lower limits are equal, then the integral is zero. k f x dx k f x dx Constant multiples can be b a moved outside. 1. 2. a a b a 3. 4. b a f x dx f x dx a Reversing the limits changes the sign. b f x dx 0 If the upper and lower limits are equal, then the integral is zero. k f x dx k f x dx Constant multiples can be b a b a moved outside. f x g x dx f x dx g x dx a a b b Integrals can be added and subtracted. 4. b a f x g x dx f x dx g x dx a a b b Integrals can be added and subtracted. 5. f x dx f x dx f x dx b c c a b a y f x a Intervals can be added (or subtracted.) b c The average value of a function is the value that would give the same area if the function was a constant: A 5 3 0 1 2 x dx 2 4 3 27 1 3 x 6 6 0 3 9 4.5 2 2 1.5 4.5 Average Value 1.5 3 1 0 1 y 2 1 2 x 2 3 Area 1 b Average Value f x dx a Width b a 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 a, b then at some point c in a, b , 1 b f c f x dx a ba p