Answer
$\dfrac{172,704}{5,632,705}\sqrt 2(1-e^{-14\pi})$
Work Step by Step
Here, $ds=\sqrt{(dx)^2+(dy)^2+(dz)^2}$
or, $ds=\sqrt{(dx/dt)^2+(dy/dt)^2+(dz/dt)^2}$
Thus, $ds=\sqrt{(-e^{-t}\cos 4t-4e^{-t} \sin 4t)^2+(-e^{-t}\sin 4t-4e^{-t} \cos 4t)^2+(-e^{-t})^2}dt=e^{-t} \sqrt {18} dt$
Now, $\int_{C} x^2y^2zds=\int_{0}^{2 \pi} (-e^{-t}\cos 4t)^3 (e^{-t} \sqrt {18} dt)$
By using a calculator, we have $\int_{C} x^2y^2zds=\dfrac{172,704}{5,632,705}\sqrt 2(1-e^{-14\pi})$
