Let $\displaystyle F=[2 \sin (x) \cos (x), 0, 2z]$ and $\displaystyle r(t)=[\cos (t),\sin (t),t]$
Calculate $\displaystyle \int_0^{2\pi} F(r(t)).r(t)dt$
I'm not sure your notation makes sense as it is. $\displaystyle F$ is a function of two(?) (real?) variables taking values in $\displaystyle \mathbb(R)^3$, and $\displaystyle r$ is a function of a single (real?) variable taking values in $\displaystyle \mathbb(R)^3$, so $\displaystyle F(r(t))$ is not properly defined. I expect with a bit of effort we could make some sort of guess about what this is supposed to denote but it would be better if you sorted it out.
CB
I figured that F(r(t)) could be worked out by using the vector format and taking xi,yj,zk to equal the values of the three points of r(t) taking the y value as 0y, and then integrate the dot product between this value and r(t), which is what i am stuck on but the question i have been set is written exactly as i have put on here.
In textbooks i have used, F seems to be denoted by F(r), but i don't know if thats actually a completely separate value!