01:640:151 Lecture Notes - Lecture 23: Integral, Antiderivative, Power Rule
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Using additivity to evaluate x(cid:2870) dx (cid:2875)(cid:2872) If m f(x) m for all x in [a, b] then m (b a) f(cid:4666)x(cid:4667) dx. Upper & lower bounds y = m y = m m (b a) a. Example: prove that (cid:885)(cid:886) (cid:2869)(cid:2934)(cid:2870)(cid:3117)(cid:3118) dx 3 y = 2 y = (cid:883)(cid:884) (cid:883)(cid:884) y = (cid:883)x . M = 2 m (b a) f(cid:4666)x(cid:4667) dx. M(b a) (cid:2912)(cid:2911) m = (cid:883)(cid:884) . A = (cid:885)(cid:887) (cid:883)(cid:884) ((cid:885)(cid:884) ) (cid:2869)(cid:2934)(cid:2870)(cid:3117)(cid:3118) dx 2 ((cid:885)(cid:884) ) (cid:885)(cid:886) . Prove that (cid:2869)(cid:2934)(cid:3118)(cid:2872)(cid:2869) x2 > x on [1, 4] dx (cid:2869)(cid:2934)(cid:2872)(cid:2869) dx (cid:2869)(cid:2934)(cid:3118) < (cid:2869)(cid:2934) on [1, 4] An antiderivative of f is a function whose derivative is f. we typically wirte capital f to denote an antiderivative of f. 2 properties of antiderivative: if f has an antiderivative, then it has infinitely many of them, all of the antiderivatives differ from each other by a constant.
