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Unformatted text preview: DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Chapter 2 APPLICATIONS OF INTEGRALS I. AREAS BETWEEN CURVES The area A of the region bounded by the curves y=f(x) and y=g(x), and the lines x=a, x=b, where f and g are continuous and f(x)≥g(x) for all x in [a,b], is [ ] ( ) ( ) . b a A f x g x dx = DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Example (i) Find the area of the region bounded above by , bounded below by and bounded on the sides by x=0 and x=1. Solution: The region is bounded by the upper curve and the lower curve x y e = y x = x y e = . y x = ( 29 1 1 2 1 2 1 1 1.5 2 x x A e x dx e x e e a = = W W = = DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II (ii) Find the area of the region enclosed by the parabolas and Solution: We first find the points of intersection of the parabolas: The points of intersection are (0,1) and (1,1). 2 y x = 2 2 y x x = 2 2 2 0 or 1. x x x x x = = = DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II The area is 1 1 2 2 2 1 2 3 (2 ) 2 ( ) 1 1 1 2 2 2 3 2 3 3 A x x x dx x x dx x x = = = = = DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II If we are asked to find the area between the curves y=f(x) and y=g(x) where f(x)≥g(x) for some values of x but g(x)≥f(x) for other values of x, then we split the given region S into several regions We then define the area of S to be the sum 1 2 1 2 , , with areas , , S S A A L L 1 2 A A A = + + L DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Since we have the following: The area between the curves y=f(x) and y=g(x) and between x=a and x=b is ( ) ( ) when ( ) ( ) ( ) ( ) ( ) ( ) when ( ) ( ), f x g x f x g x f x g x g x f x f x g x = ( ) ( ) b a A f x g x dx = DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Example Find the area of the region bounded by the curves Solution: The points of intersection occur when We see and sin , cos , 0, / 2. y x y x x x π = = = = sin cos / 4. x x x π = = cos sin when 0 / 4 x x x π cos sin when / 4 / 2. x x x π π DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II [ ] [ ] / 2 1 2 / 4 / 2 / 4 / 4 / 2 / 4 cos sin (cos sin ) (sin cos ) sin cos cos sin 1 1 1 1 0 1 0 1 2 2 2 2 2 2 2. A x x dx A A x x dx x x dx x x x x π π π π π π π = = + = + = + +  = + +   + + = DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Note Some regions are best treated by regarding x as a function of y . If a region bounded by curves with equation x=f(y), x=g(y), y=c, and y=d, where f and g are continuous on [c,d], then ( ) ( ) ....
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This note was uploaded on 04/29/2009 for the course EE 307 taught by Professor Dinh during the Spring '09 term at Washington State University .
 Spring '09
 Dinh

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