Volumes - 2 2 2 2 1 4 1 4 1 4 1-=-⇔-=-= y x y y x y y x...

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6.2 Volumes If a region in the plane is revolved about a line, the resulting solid is called a solid of revolution , the line is called the axis of revolution . A. The Disc Method B. The Washer Method the strip is perpendicular to the axis of revolution
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A. Volume: The Disc Method Axis of Revolution x = a x = b Plane Region Vertical Strip x r Typical disc x Solid of Revolution x r V = 2 π Volume of Typical Disc where r = radius Volume of Solid ( 29 [ ] ( 29 [ ] dx x r x x r V b a n i i n 2 2 1 lim = = = π π
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B. Volume: The Washer Method a b Axis of Revolution Typical Washer 0 r i r Vertical Strip Plane Region x x Solid of Revolution Volume of Typical Washer ( 29 x r r V i - = 2 2 0 π where 0 r i r = outer radius = inner radius Volume of Solid [ ] [ ] ( 29 - = b a i dx x r x r V 2 2 0 ) ( ) ( π
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Examples Find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disc or washer. Example One: ; 0 , 2 = - = x y y x about the y -axis Example Two: ; 3 , 1 , 0 , 1 = = = = x x y x y about 1 - = y
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Solutions: Example One
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Unformatted text preview: 2 2 2 2 1 4 1 4 1 4 1 --=-⇔ +--=--= y x y y x y y x Graph is a parabola with vertex of (1/4,1/2) (1/4,1/2) 1 x y y ∆ r 2 y y x r-= = y x ( 29 30 5 1 2 1 3 1 5 2 3 ) 2 ( 1 5 4 3 1 4 3 2 2 1 2 π = +-= +-= +-=-= ∫ ∫ y y y dy y y y dy y y V By disc method Solutions: Example Two x y 1 = 1-= y x y 1 3 x ∆ r i r 1-= y x y By washer method ( 29 1 1 1 1 1 ) 1 ( =--= + = + =--= i r x y y r ( 29 ( 29 + = +-- +-= +-= + = - + = ∫ ∫ 3 1 3 ln 2 1 3 ln 2 3 1 ln 2 1 2 1 1 1 1 3 1 3 1 2 3 1 2 2 π x x dx x x dx x V...
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