Which of the following integrals represents the length of the parametric curve $x=1+e^t,\;\;y=t^2,\;\;-3 \le t \le 3,$ about the $x$-axis?

A. $\displaystyle \int_{-3}^{\,3} {\sqrt{e^{t^2}+4t^2}}\,dt$
B. $\displaystyle \int_{-3}^{\,3} {\sqrt{e^{t^2}+4t}}\,dt$
C. $\displaystyle \int_{-3}^{\,3} {\sqrt{e^{2t}+4t^2}}\,dt$
D. $\displaystyle \int_{-3}^{\,3} {\sqrt{e^{2t}+4t}}\,dt$
E. $\displaystyle \int_{-3}^{\,3} {\sqrt{2e^{t}+4t^2}}\,dt$
F. $\displaystyle \int_{-3}^{\,3} {\sqrt{2e^{t}+4t}}\,dt$