[Find the work done by the force $ \boldsymbol{F}=-10y\boldsymbol{i}+4x\boldsymbol{j}$ along one loop of the curve $r=\sin(9\theta)$]
I have some trouble formatting/understanding the question.
I have it set up as something like
$\int_c\boldsymbol{F}(\boldsymbol{r}(t))\cdot \boldsymbol{r}'(t)dt$
Where the $x, y$ components of F are rewritten with respect to $r$ (which is equal to $\sin(9\theta)$) and $\theta$.
$ \boldsymbol{F}=-10\sin(9\theta)\sin(\theta)\boldsymbol{i}+4\sin(9\theta)\cos(\theta)\boldsymbol{j}$
with parameterization
$ \boldsymbol{r}(t)=\sin(9\theta)\cos(\theta)\boldsymbol{i}+\sin(9\theta)\sin(\theta)\boldsymbol{j}$
And then I go about to find the derivative of $\boldsymbol{r}(t)$, find the dot product, and putting everything together I integrate with respect to $\theta$ from $0$ to $2\pi$.
It gets really messy, and im pretty sure this is suppose to be an easier question. Am I doing something wrong or is there an easier way to go around solving this question?