Find the arc length on the interval for t between 0 and 2 inclusive for the curve

Question

Find the arc length on the interval for t between 0 and 2 inclusive for the curve described with the parametric equations: x = 3t², y = 2t³ + 1. (4 points)

answer choices

Answer 1

S(t) = ∫√((dx/dt)² + (dy/dt)²) dt integrate from 0 to 2

dx/dt = 6t and dy/dt = 6t²

S(t) = ∫√(36t² + 36t⁴) dt = 6∫t√(1+t²)dt from 0 to 2

let u = √(1+t²)

S(u) = 6∫u²du integrating from 1 to √5

= 6u³/3 from 1 to √5 = 2(5√5 - 1) = 10√5-2

Which is the 4th selection

Answer 2

The correct answer is the last one.

arc_length = ∫ dt sqrt[ (dx/dt)² + (dy/dt)² ] { limits 0 to 2 }

This leads to 6 ∫ t sqrt ( 1 + t² ) dt

The anit-derivative can be worked out using the u substitution method.