2311lectureoutline5.2

2311lectureoutline5.2 - 1) 6 n i n n n i = + + = 2 2 3 1 (...

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MAC2311, Calculus I 5.2 Areas, Riemann Sums and the Definite Integral Definition of a Definite Integral: Let a function f(x) be defined on the interval [a, b], and the interval is divided into n subintervals of equal width b a x n - Δ = . Then the definite integral of f from a to b is: 1 ( ) lim ( ) b a n i i n i f x dx f x x →∞ = = Δ ± where b a x n - Δ = and i x a i x = + Δ , provided the limit exists. This definite integral gives us “signed area” between the curve and the x-axis. Any area above the x-axis is positive and any areas below the x-axis are negative. Summation Properties & Formulas: 1 1 1 ( ) n n n i i i i i i i a b a b = = = ± = ± ± ± ± 1 1 n n i i i i ca c a = = = ± ± , where c is a constant 1 n i c cn = = ± 1 ( 1) 2 n i n n i = + = ± 2 1 ( 1)(2
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Unformatted text preview: 1) 6 n i n n n i = + + = 2 2 3 1 ( 1) 4 n i n n i = + = Properties of the Definite Integral: ( ) ( ) a b b a f x dx f x dx = -& & ( ) a a f x dx = & ) ( ( ) ( ) ( ) ( ) f x g x dx f x dx g x dx = & & & ( ) ( ) b b a a c f x dx cf x dx = & & (c is a constant) Examples: Find the area of the region bounded by f(x) = x 3 , the x-axis, x = 0 and x = 1, using the limit definition. Find the area of the region bounded by f(x) = 10 x 2 , the x-axis, x = 2 and x = 3, using the limit definition. Try exercises 2, 6, 18, 20, 22, 24, 30, 34, 36, 38, 48 on pages 376-378 in your text....
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