Problem: A force field is given in polar coordinates by the equation $$F(r,\theta)= (-4\sin (\theta), 4\sin (\theta)).$$
Compute the work done in moving a particle from the point $(1,0)$ to the origin along the spiral whose polar equation is $r = e^{-\theta}$.
my Try:
what I think is to take $\alpha(t)=(e^{-t},t)$ $ ( \alpha'(t)=(-e^{-t},1))$ to describe the curve but $0 \leq t \leq +\infty$ beacuse $\lim_{t\rightarrow +\infty} e^{-t}=0$ and that's the origin.
then $\int^{+\infty}_{0} F(\alpha(t))\cdot \alpha'(t) \,dt=\int^{+\infty}_{0} \sin(t)e^{-t}+\sin(t)\, dt$ and the last doesn't converge. I think my problem is do not parameterize the curve in a good way.