Calculate the length of the path over the given interval: c(t) = (3t^2, 4t^3), 2<t<5

Answer 1

s = ∫[2,5] √(dx^2+dy^2) dt
= ∫[2,5] √((6t)^2 + (12t^2)^2) dt
= ∫[2,5] √(36t^2 + 144t^4) dt
= ∫[2,5] 6t√(1+4t^2) dt
Letting u = 1+4t^2, we have
= ∫[17,101] 3/4 √u du
= (3/4)(2/3) u^(3/2) [17,101]
= 1/2 (101√101 - 17√17)