How Do Parametric Curves and u-Substitution Relate to Length Calculation?

[bobsmith76]

Homework Statement

x = t^3, y = (3t^2)/2 0<= t <= √3

The Attempt at a Solution

dx/dt = 3t^2
dy/dt = 3t

step 1. √((3t^2)^2 + (3t)^2)

step 2. 3t^2 + 3t

(the book says I can't do that, I don't see why)

step 3. insert √3 into t

Here's the books solution

I understand u substitution but I don't see why it's necessary here. I don't why my method does not follow all the rules.

 

[vela]

Because $\sqrt{a^2+b^2} \ne a+b$. Try it with a=b=1, for instance.