Why dot in $W = \int \mathbf F\cdot \mathrm d\mathbf s$?

Question essentially in title. Why do we use the dot symbol when writing $$W = \int \mathbf F\cdot \mathrm d\mathbf s$$? I understand that $$\mathbf F$$ and $$\mathrm d\mathbf s$$ are vectors and that probably has something to do with it, but when I read this I don't see "the integral of $$\mathbf F$$ with respect to $$\mathrm d\mathbf s$$", I see "find $$\mathbf F\cdot \mathrm d\mathbf s$$, and THEN take the integral of that". It's kind of tripping me up, and I'm just wondering why the dot is there?

Consider that $$\vec{ds} = \hat{r}ds$$ where $$\hat{r}$$ is a unit vector pointing along the path of the object, and $$ds$$ is just the regular differential line element. Now also note that at every point $$p$$ along the path you can separate the component of the force that's along the path from the component perpendicular to the path: $$\vec{F}(p) = F_\parallel(p)\cdot\hat{r} + F_\perp(p)\cdot\hat{r}_{\perp}$$
Where $$\hat{r}_{\perp}$$ is just a unit vector along the axis perpendicular to the path. So the job of $$\vec{ds}$$ is simply to pick the component that's along the path at every point. In other words at a point $$p$$ again: $$\vec{F}(p)\cdot\vec{ds} = \vec{F}(p) \cdot \hat{r}(p)ds = F_{\parallel}(p) ds$$
Note that at every point $$F_{\parallel}(p)$$ may be either positive, zero or negative. I had the magnitude of the parallel component $$|\vec{F_{\parallel}}(p)|$$ before this edit as the result, which is incorrect! The force may also be directed opposite to the path and this should contribute negatively to the overall work.
$$\mathbf{F}\cdot d\mathbf{s}$$ is the dot product of the vectors $$\mathbf{F}$$ and $$d\mathbf{s}$$. The result is a scalar, in this case the work $$dW$$. Its geometric definition is $$\mathbf{F}\cdot d\mathbf{s} = ||\mathbf{F}||\ ||d\mathbf{s}||\ \cos\theta$$ where $$||\mathbf{F}||$$ and $$||d\mathbf{s}||$$ are the magnitudes of the vectors $$\mathbf{F}$$ and $$d\mathbf{s}$$, and $$\theta$$ is the angle between these two vectors.