What is the length of the plane curve defined by r(t) = <t^2 + 1, t^3 - 2> for t is an element of [0,2]?
So would it be $\displaystyle
\int_{a}^{b}\sqrt{(2t)^{2}+(3t^2)^{2}}dt
$
$\displaystyle
\int_{a}^{b}\sqrt{4t^{2}+9t^4}dt
$
$\displaystyle
\int_{a}^{b}t\sqrt{4+9t^2}dt
$
$\displaystyle
u = {4+9t^2}du
$
and $\displaystyle
du= 18t
$
So $\displaystyle
(1/18)\int_{a}^{b}\sqrt{u}du
$
$\displaystyle
(1/27)(u)^(3/2)
$
$\displaystyle
(1/27)(4+9b^2)^(3/2)-(1/27)(4+9a^2)^(3/2)
$
Is this the right answer?