So I have the following exercise:
Student is running in the straight corridor with the spdeed $v = \frac{63}{200}t\cos t$ m/s where $t$ is the time that has passed since the student started running in seconds. Find the distance that student has traveled during the period of time $ t \in [6,11]$ , the distance form the starting point and average speed.
What I found out is that student is $-3.24$ m from the starting point, but I can't find the total distance that he has travelled, since I can't put that $\frac{63}{200}t\cos t =0$ and $\int|\frac{63}{200}t\cos t |$ doesn't exist either. And I also don't know the equation for finding the average speed
The average of a function $f:\mathbb{R}\to\mathbb{R}$ on $[a,b]\subseteq\text{dom}(f)$ is $$\bar f_{[a,b]} = \frac{1}{b-a}\int^b_af$$ And by definition, $$v=\dot{x}$$ So total distance travelled will be $$d=\int_a^b|v|\mathrm{d}t$$ It's not so hard to find the roots of your velocity function, and based on that, you can take apart the integral into $3$ smaller one, and remove the absolute value sign on them accordingly to the sign of the velocity.