Ex 2 : a Find the length of the...

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Further Applications of Integration Arc Length, Surface Area, Parametric Curves (8.1) Arc Length If f 0 is continuous on [ a, b ] , then the length of the curve y = f ( x ) , a x b , is L = Z b a p 1 + [ f 0 ( x )] 2 dx = Z b a s 1 + dy dx 2 dx. Ex 1 : Find the length of the curve y = 2 - x 2 , 0 x 1 . (8.1 #2) 1
The Arc Length Function Let y = f ( x ) be a smooth curve on a x b and let the function s ( x ) be the distance along the curve y = f ( x ) from the initial point ( a, f ( a )) to the point ( x, f ( x )) . Then s ( x ) is defined by the formula s ( x ) = Z x a q 1 + [ f 0 ( t )] 2 dt. Observe that ds dx = So we can rewrite the arc length formula as L = Z ds = Z s 1 + dy dx 2 dx = Z s dx dy 2 + 1 dy 2
Ex 2 : (a) Find the length of the curve x = y 4 y 2 , 1 y 2 . 4 8 + 1 (8.1 #10) (b) Find the arc length function s ( y ) for the curve x = y 4 8 + 1 4 y 2 from the initial point ( 3 8 , 1) to ( g ( y ) , y ) , for y > 0 . 3
(8.2) Area of a Surface of Revolution If f is positive and f 0 is continuous on a x b , we de- fine the surface area of the surface obtained by rotating the curve y = f ( x ) , a x b , about the x -axis, as S = Z b a 2 πf ( x ) p 1 + [ f 0 ( x )] 2 dx = Z b a 2 πy s 1 +