dx F b F a p 24 Definite Integrals Example Evaluate integraldisplay 1 1 x 2dxby

Dx f b f a p 24 definite integrals example evaluate

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dx = F ( b ) F ( a ) . – p. 24/ ??
Definite Integrals Example Evaluate integraldisplay 1 0 (1 - x 2 ) dx by the Fundamental Theorem of Calculus. integraldisplay 1 0 (1 - x 2 ) dx = bracketleftBig x - 1 3 x 3 bracketrightBig 1 0 = 1 - 1 3 = 2 3 0 1 0 1 y = 1-x 2 Recall that we computed and got the same area under the graph of y = 1 x 2 by the limiting process via Riemann sums earlier. – p. 25/ ??
(iv)u(x) = cosx,0xπ/2.
Definite Integrals Example Evaluate the following definite integrals. (i) integraldisplay 2
Remark - Natural Log One could define the natural logarith- mic function by a definite integral: ln x def = integraldisplay x 1 1 t dt for any x > 0 , i.e., ln x is consid- ered as some area function under the graph of y = 1 x . 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 y = 1/x y x integraldisplay 2 1 1 t dt def = ln2 In particular, e is the number chosen to get a unit area integraldisplay e 1 1 t dt = 1 . All basic properties of the natural logarithmic function ln x could be derived from the integral, and the natural exponential function e x could be defined by inverting ln x . Laws of Exponents could be derived from those basic properties of ln . – p. 28/ ??

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