Areas - -=-=-=- y y y dy y y dy y y A y ∆ Solutions...

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6.1 Areas Between Curves Suppose we want to find the area of the region S as shown in Figure 1. f g x y b a Figure 1 S
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f g x y b a [ ] [ ] dx x g x f x x g x f A b a n i i i n ) ( ) ( ) ( ) ( lim 1 - = - = = ) ( ) ( i i x g x f - x i x Figure 2
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When the graph of the area of the region is already determined, it is often convenient to use a typical approximating rectangle (strip) to write the integration formula for the area between two curves. In general, we use ( 29 ( 29 [ ] - = 2 1 curve bottom curve top x x dx A Vertical Strips (of width ) implies integration with respect to x x Horizontal Strips (of width ) implies integration with respect to y y in variable x ( 29 ( 29 [ ] - = 2 1 curve left curve right y y dy A in variable y
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Examples Sketch the region bounded by the given curves and find the exact area of the region. Example One: Example Two: 1 , 3 2 = - - = y x x y 2 / , 0 , 2 sin , sin π = = = = x x x y x y
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Solutions: Example One Find the points of intersection of the graphs: 1 , 3 2 = - - = y x x y ( 29 ( 29 2 or 1 2 1 0 2 0 1 3 2 2 - = = + - = - + = + = - y y y y y y y y (2,1) (-1, -2) x y 1 + = y x 2 3 y x - = The area is: ( 29 ( 29 [ ] ( 29 2 9 4 2 3 8 2 2 1 3 1 2 2 3 2 1 3 1 2 1 2 2 3 2 1 2 2 = - - - + - - =
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Unformatted text preview: +--= +--= +--=---∫ ∫ y y y dy y y dy y y A y ∆ Solutions: Example Two Find the points of intersection of the two curves: x y x y 2 sin , sin = = 2 / ; 3 / or , 2 1 cos or , sin ) 1 cos 2 ( sin sin cos sin 2 cos sin 2 sin 2 sin sin π ≤ ≤ = = = = ⇒-=-= = = x x x x x x x x x x x x x x x 2 / 3 / 1 ( 29 2 / 3 , 3 / x y x y 2 sin = x y sin = The total area is: 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 cos 2 1 cos cos 2 cos 2 1 ) 2 sin (sin ) sin 2 (sin 2 / 3 / 3 / 3 / 2 / 3 / = -+-- --+ +-- + --= +-+ +-=-+-= ∫ ∫ x x x x dx x x dx x x A x ∆ x ∆...
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