Find the arc length of the graph of the function over the indicated interval.

Question

x=1/3(x)^1/2(y − 3),    1 ≤ y ≤ 16

Answer 1

S = Integral of ds, which is sqrt(dx²+dy²) or, changing to integrate in terms of y:

S = Integral from 1 to 16 of sqrt((dx/dy)² + 1) dy

simplifying the equation (and staying away from x = 0 : x = 1/9 (y-3)²

S = integral from 1 to 16 of ((2/9(y-3))² + 1)^1/2 dy

You can numerically integrate this or do an atanθ substitution.