Find the length of the arc of the parabola $y^2=4x$ from $x=0$ to $x=4$.
In the manual solution is $2\sqrt{5}+\ln(2+\sqrt{5}).$
My answer is
$\displaystyle 2\int_0^4 \! \sqrt{1+\frac{\mathrm{d}x}{\mathrm{d}y}} \, \mathrm{d}x$
$\displaystyle 2\int_0^4 \! \sqrt{1+\frac{y^2}{4}} \, \mathrm{d}x$
$\displaystyle 2\int_0^4 \! \sqrt{4+y^2} \, \mathrm{d}x= 4 \sqrt{5}+2\ln(2+\sqrt{5})$
But my answer is $4 \sqrt{5}+2\ln(2+\sqrt{5})$ As I doubled the length, Do we need to double the arc length?