2-16 - dx dt ² 2 ± dy dt ² 2 and ds dt = r ³ x t ´ 2...

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February 16, 2005 Announcements Midterm # 2 is next week! Thursday, February 24th (Covers § 6.4 - 7.8) Start doing practice Midterms. Modification to Week 8 homework: Problems 1, 5, 7, 11, 12 and 15 in § 8.3 are NOT part of the assigned homework. Today § 7.8 More on Improper Integrals: A couple more examples § 8.1 Arc Length: What does length of a curve mean? How do you compute it? 1
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The arc length formula If f is continuous on [ a, b ], then the length of the curve { ( x, y ) : a x b, y = f ( x ) } , is L = b a 1 + [ f ( x )] 2 dx, or L = b a 1 + dy dx 2 dx. The arc length formula If g is continuous on [ c, d ], then the length of the curve { ( x, y ) : c y d, x = g ( y ) } , is L = d c 1 + [ g ( y )] 2 dy, or L = d c 1 + dx dy 2 dy.
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The arc length function s of the curve C = { ( x, y ) : a x b, y = f ( x ) } is the distance along the curve C from the ini- tial point P 0 = ( a, f ( a )) to Q = ( x, f ( x )), i.e. s ( x ) = x a 1 + [ f ( t )] 2 dt. By the FTC ds dx = 1 + [ f ( x )] 2 = 1 + dy dx 2 . Thus (1) ( ds ) 2 = ( dx ) 2 + ( dy ) 2 . Symbolically L = ds.
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This is very useful when the curve C is given parametrically by C = { ( x ( t ) , y ( t )) : t 0 t t 1 } . Then (1)
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Unformatted text preview: dx dt ² 2 + ± dy dt ² 2 . and ds dt = r ³ x ( t ) ´ 2 + ³ y ( t ) ´ 2 . In this case L = Z t 1 t ds dt dt = Z t 1 t r ³ x ( t ) ´ 2 + ³ y ( t ) ´ 2 dt. Problem § 8.1 # 34. A steady wind blows a kite due west. The kite’s height above the ground from horizontal position x = 0 to x = 80 ft is given by y = 150-1 40 ( x-50) 2 . Find the distance traveled by the kite. Problem § 8.1 # 33. A hawk flying at 15m/s at an altitude if 180m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y = 180-x 2 45 until it hits the ground. Here y is the height above the ground and x the horizontal distance traveled in meters. Calculate the distance trav-eled by the prey from the time it is dropped until the time it hits the ground....
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