Length of A Curve

Exercises 1.1
11 (a - c)



Note: If this question is too advanced for me right now in my studies, please skip.

 

Consider the length of the graph of f(x) = 5/x from (1, 5) to (5, 1).
(a) Approximate the length of the curve by finding the distance between its two endpoints.

the length of the graph is equal to the distance between (1, 5) and (5, 1)

d=sqrt((5-1)^2+(1-5)^2)
d=sqrt(4^2+(-4)^2)
d=sqrt(16+16)
d=sqrt(32)
d=4sqrt(2) ≈5.65685
(b) Approximate the length of the curve by finding the sum of the lengths of four line segments.
only way to use distance formula for each segment is to use f(x) = 5/x and find coordinates of each point that has x coordinates 2,3, and 4
f(2) = 5/2 -> point is (2 , 5/2)
f(3) = 5/3 -> point is (3 , 5/3)
f(4) = 5/4 -> point is (4 , 5/4)

distance between (1, 5) and (2, 5/2)=sqrt((2-1)^2+(5/2-1)^2)=sqrt(13)/2
distance between (2, 5/2) and (3 , 5/3) =sqrt((3-2)^2+(5/3-5/2)^2)=sqrt(61)/6
distance between (3 , 5/3) and (4 , 5/4) =sqrt((4-3)^2+(5/4-5/3)^2)=13/12
distance between (4 , 5/4) and (5, 1)=sqrt((5-4)^2+(1-5/4)^2)=sqrt(17)/4

add all:
sqrt(13)/2+sqrt(61)/6+13/12+sqrt(17)/4≈5.218593

(c) Describe how you could continue this process to obtain a more accurate approximation of the length of the curve.

I would find more points on the curve, as shorter the length of segment is as better approximation will be

 

Great study notes. Thank you very much.