Find the arc length of
$$r\langle t\rangle = \langle e^{t^2},3e^{t^2}-2,\frac{3}{2}e^{t^2}\rangle\text{ for } 0\le t\le 1$$
My try:
Arc length $= \int | r^1 (t) | \, dt$
$$ r^1(t) = \langle 2te^{t^2}, 6te^{t^2}, 3te^{t^2} \rangle $$
$$ |r^1(t)| = \sqrt{49t^2 e^{2t^2}} = 7te^{t^2}$$
Arc length $\displaystyle = \int_0^1 7te^{t^2} \, dt = \dfrac 7 2 (e - 1)$
Is my above attempt correct? Can anyone please verify.