MATA37H3 Lecture Notes - Lecture 3: Jean Gaston Darboux, Itz, Riemann Sum
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Mata37 - lecture 3 - de nite integrals continued. Properties of the de nite integral (chapter 4. 3: see pages 344, 346, 347 and supplements, let a, b r, a < b. Suppose f, g are integrable on [a, b]. Then: if f (x) 0 x [a, b], then. 6. f (x)dx = 0 f (x)dx = f (x)dx (cid:90) a (cid:90) c b f (x)dx = f (x)dx + a a (see diagram 1) Want to show (f (x) g(x))dx = f (x)dx (cid:90) b (cid:90) b a g(x)dx (cid:90) b (cid:90) b a a a f (x)dx = lim n g(x)dx = lim n n(cid:88) n(cid:88) i=1 i=1. Consider right hand side: by given info a f (x i ) x, and g(x (cid:90) b (cid:90) b f (x)dx a a. Qed: example #1: let g(x)dx = 10. 1 x2dx, by interval properties g(x)dx +
