How do you solve?

Question

What is the length of the curve x = y^3/2 from y = 0 to y = 4? (4 points)

A) None of the choices below

B)

C)

D)

Answer 1

Here y = c at Point A and y = d at Point B, with dx/dy equal to 1.5y^0.5.

From s = ∫_{(from A to B)}ds equal to ∫_{(from c to d)} [1 + (dx/dy)²]^0.5 dy,

construct Arc Length = ∫_{(from 0 to 4)} [1 + (1.5y^0.5)²]^0.5 dy.

Rewrite as ∫_{(from 0 to 4)} [1 + 2.25y]^0.5 dy.

Integrate to (4/9)(2/3)[1 + (9/4)y]^1.5.

Reduce to (8/27){[1 + (9/4)y]^1.5|_{(from y = 0 to y = 4)}}.

Evaluate to (8/27){[1 + (9/4)4]^1.5 - [1 + (9/4)0]^1.5}.

Simplify to (8/27)[10√10 − 1] or Choice D.