121616949-math.240 - x i and the surface area integral...

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226 Chapter 9 Applications of Integration f is - x/ p r 2 - x 2 , so the surface area is given by A = 2 π Z r - r p r 2 - x 2 r 1 + x 2 r 2 - x 2 dx = 2 π Z r - r p r 2 - x 2 r r 2 r 2 - x 2 dx = 2 π Z r - r r dx = 2 πr Z r - r 1 dx = 4 πr 2 If the curve is rotated around the y axis, the formula is nearly identical, because the length of the line segment we use to approximate a portion of the curve doesn’t change. Instead of the radius f x i ), we use the new radius ¯
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Unformatted text preview: x i , and the surface area integral becomes Z b a 2 πx p 1 + ( f ± ( x )) 2 dx. EXAMPLE 9.39 Compute the area of the surface formed when f ( x ) = x 2 between 0 and 2 is rotated around the y-axis. We compute f ± ( x ) = 2 x , and then 2 π Z 2 x p 1 + 4 x 2 dx = π 6 (17 3 / 2-1) , by a simple substitution....
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