This preview shows pages 1–10. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 = = = = sin cos / 4. x x x = = cos sin when 0 / 4 x x x cos sin when / 4 / 2. x x x 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 = = + = + = + +  = + +   + + = 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 ( ) ( ) ....
View Full
Document
 Spring '09
 Dinh

Click to edit the document details