Find the length of the curve

Question

I need to find the length of the curve $$c(t)=(3e^{t}-3, 4e^{t}+7)$$ for $$0\le t \le 1$$

If I understand correctly, I need to take the derivative of the y part of that coordinate over the derivative of the x of that coordinate and simplify and then do something else?

Answer 1

The lenght of a curve $\gamma: [a,b] \to \mathbb{R}^n$ is given by

$$ Lenght(\gamma) = \int\limits_a^b || \gamma'(t) || dt $$

As in your case, $c'(t) = ( 3e^t, 4e^t ) $ and $||c'(t)|| = \sqrt{ 25 e^{2t} } = 5 e^t$

Now, you should conclude.