Answer
$\frac{136 \sqrt{17}-128 \sqrt{2}}{27}\approx 14.1$
Work Step by Step
We are given the vector:
$r(t)=\lt 2t^2+1,2t^2-1,t^3 \gt$
Recall that $(x^n)'=nx^{n-1}$
Thus, we get the vector derivative:
$r'(t) =\lt 4t, 4t, 3t^2\gt$.
We calculate the length by integration: $$length=\int_{0}^{2}\|r'(t)\|dt=\int_{0}^{2}\sqrt{16t^2+16t^2+9t^4} dt\\=\frac{1}{18}\int_{0}^{2}18t\sqrt{32 +9t^2} dt=\frac{2}{18(3)}(32+9t^2)^{3/2}|_{0}^{2}\\=\frac{1}{27}((68)^{3/2}-(32)^{3/2})=\frac{136 \sqrt{17}-128 \sqrt{2}}{27}\approx 14.1.$$
