Solution
= Solution
Differentiate the parametrized <curve> to obtain $\boldsymbol x'(t)=(1-t^2,2t,1+t^2)$. Its <speed> obeys
$$
|\boldsymbol x'(t)|^2=(1-t^2)^2+4t^2+(1+t^2)^2=2(1+t^2)^2.
$$
Since $1+t^2>0$, the <arc length> is
$$
\boxed{L=\int_0^1\sqrt2(1+t^2)\,dt=\frac{4\sqrt2}{3}.}
$$