Average mechanical power formula wrong?

Question

In my physics book it states that

Average mechanical power is the ratio between the work $W$ done by a force during a time interval $\Delta{t}$ and the time interval. Hence $$P_{ave}=\frac{W}{\Delta{t}}=\vec{F}\cdot\vec{v_{ave}}$$

I do not agree with this statement and I think this is only the case for constant direction of motion and constant force, since if the force is not constant, it is unclear what force is used in the formula.

$$\vec{F}\cdot\vec{v_{ave}}=\vec{F}\cdot\frac{\Delta{\vec{r}}}{\Delta{t}}$$

The next step would be to say that

$$\vec{F}\cdot\Delta{\vec{r}}=W$$

but as of what I know this is only the case for constant direction. For non uniform paths

$$W=\int\limits_{\vec{r}=\vec{r_i}}^{\vec{r}=\vec{r_f}} \vec{F}\cdot d\vec{r}$$

Can anyone please explain this to me? I am confused... Who is wrong, me or the book?

Answer 1

You are correct. You might want to check the context or limitations for the formula from the book.

Answer 2

If $\vec{F}$ is $\vec{r}$-independent, it can be moved outside your integral as a coefficient, so the formula you were told is an important special case of the more general result you've obtained.