Question

use a grapher to fine the length numerically (x)=2sin (y) ,pi/3 < y< 2pi/3

Answer 1

Since I cannot draw a graph for you, I would break the interval up into 6 intervals of length pi/18. Calculate the change in y = f(x) for each interval and then compute sqrt(deltay^2 + deltax^2)as the length of a straight line connecting the points, delta-s. Then add them up the "delta-s" line elements. Use a table.

x.....f(x)..delta y .. delta x.. delta s
pi/3..1.7321.------.....------.....0
7pi/18.1.8794...0.1474...0.1745..0.2284
8pi/18.1.9696...0.0902...0.1745..0.1964
9pi/18.2.0000...0.0304...0.1745..0.1771
10pi/18.1.9696..-0.0304..0.1745..0.1771
11pi/18 1.8794..-0.0902..0.1745..0.1964
12pi/18 1.7321..-0.1474..0.1745..0.2284
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sum of delta s = 1.2038

This is considerably more than the change in x along the curve, which is pi/3 = 1.0472