TITLE: Why do I get two different answers when solving for arclength?
QUESTION [8 upvotes]: I am given that $\frac{dx}{dt}=8t\cos(t)$ and $\frac{dy}{dt}=8t\sin(t)$. I tried solving for the arclength from $t=0$ to $t=1.$ 
Method 1:
$$\text{Arclength} = \int_{0}^{1} \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} dx = 4.$$
Method 2:
$$\text{Arclength} = \int_{0}^{1} \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx.$$ However, when I solve using method 2, I get $1.22619,$ when the answer should be $4.$ What is causing this difference?

REPLY [3 votes]: The second method should give you the correct answer as well. 
Note that $$ \sqrt{1+(\frac{dy}{dx})^2} dx =\sqrt {1+\tan^2(t)}(8t\cos(t))dt$$
so the arc length is $$\int _0^1 \sqrt {1+\tan^2(t)}(8t\cos(t))dt = \int _0^1 8tdt=4$$