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# Length of Polygonal Line Segments and Length of a Curve and Distance Formula

Find the total length of the polygonal line segments joining the points (xi, f(xi),i=0, 1,...,n, zwhere a= x0,x1,... xn=b is a regular partition of (a,b). use the indicated values for n

(1) f(x) = sqrt x, a=0,b=4
(a) n=2, (b) n=4

(2) f(x) = sin^2 x, a=0, b= 2pi
(a) n=2 (b) n=4 (c) n=8

(3) Use a y integration to find the length of the segment of the line y= 2x+3 =o between y=1 and y=3 check by the distance formula.

Find the length of the indicated curve.

(1) y = 2/3(x^2 +1) ^ 3/2 between x=1 and x=2

(2) y = (x^4 +3) /(6x) between x=1 and x=3

(3) 30xy^3- y^8 =15 between y= 1 and y =3

#### Solution Preview

(1)&(2)
The total length of a polygonal path is the sum of the length of its line segments.
In general, if we are joining points on the graph of a function f, the total length will be the sum of the lengths of the line segments (x_(i+1), f(x_(i+1)))-(x_i,f(x_i)), i.e.
the sum over i=0,...,n-1 of
sqrt((x_(i+1)-x_i)^2 + (f(x_(i+1))-f(x_i))^2)
.

x_i , i=0,...,n ,form a regular partition of (a,b) means that x_(i+1)-x_i = (b-a)/n.

So for (1), part (a), we have:
n = 2, b=4, a=0, so x_(i+1)-x_i = 2, x_0 = 0, x_1=2, x_2=4, and the total length of ...

#### Solution Summary

Length of Polygonal Line Segments and Length of a Curve and Distance Formula are investigated.

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