Answer
$0.8208$
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, $ds=\sqrt{(1)^2+(2t)^2+(-e^{-t})^2}dt=\sqrt {1+4t^2+e^{-2t}} dt$
Now, $\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}=\int_0^1 (e^{-t})(e^{-t\cdot t^2}) (\sqrt {1+4t^2+e^{-2t}} dt)$
By using a calculator, we have: $\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}=0.8208$
