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Section5.4 - Math 2B Dr C Famiglietti Section 5.4 Arc...

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Math 2 B Dr. C. Famiglietti ************************************************************************ Section 5.4: Arc Length and Surface Area Arc Length (length of a curve) If f = df dx is continuous on a , b [ ] , then the length of the curve (i.e., arc length) y = f x ( ) on a x b is: s = 1 + f x ( ) ( ) 2 dx a b = 1 + df dx 2 dx a b . If g = dg dy is continuous on c , d [ ] , then the length of the curve (i.e., arc length) x = g y ( ) on c y d is: s = 1 + g y ( ) ( ) 2 dy c d = 1 + dg dy 2 dy c d Surface Area (of a solid of revolution) If curve y = f x ( ) , a x b , is rotated about the x-axis (with f = df dx continuous on a , b [ ] ), then the surface area of the resulting solid of revolutions is: S = 2 π f x ( ) a b 1 + f x ( ) ( ) 2 dx = 2 π f x ( ) 1 + dy dx 2 dx a b Note: The above formulas lead to difficult integrals that often have to be integrated numerically. Example. Use integration to find the length of the line segment y = 2 x + 3 between x = 1 and x = 3 . Check by the distance formula. Solution. f x ( ) = 2 x + 3 f x ( ) = 2 Therefore, s = 1 + f x ( ) ( ) 2 dx a b = 1 + 2 2 dx 1 3 = 5 dx 1 3
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= 5 x ] 1 3 = 5 3 1 ( ) = 2 5
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