CHAPTER XVIII 



142. Lengths of Curves. If P and Q are two points taken very 

 close together on a curve, the length of that part of the curve 

 between P and Q being 8s. Then 8s can be taken approximately 

 as the hypotenuse of a right-angled triangle, the base of which 

 is 8# and the perpendicular $y. The smaller 8# is made the more 

 nearly true does this approximation become, and it becomes 

 actually true when 8# is made infinitely small. 



Then 8s 2 = 8 2 + S?/ 2 



or 



and also i ^ , *. , v ., 



%/ \Sy< 



In the limit when S# is made infinitely small, 

 ds 



ds 

 or 



ay 



Thus, to get s, the length of a certain portion of the curve, 

 the first of these expressions must be integrated with respect 

 to x between assigned limits, or the second expression must be 

 integrated with respect to y between assigned limits. 



Then 



or 



Example 1. Find the length of the arc of curve of y 2 

 between the limits x = 1 and x = 3. 



y = 



268 



