Question

A bee with velocity vector r ′ ( t ) starts out at the origin at t = 0 and flies around for T seconds. Where is the bee located at time T if ∫ T 0 r ′ ( u ) d u = 0 ? What does the quantity ∫ T 0 ∥ r ′ ( u ) ∥ d u represent?

Answer 1

At time T the bee will be at (0,0).

The quantity represents the integral of function
`[r^(')(u)]` with the time interval of (0,T).

Explanation:

Given :

`\int\limits^T_0 {r^(')(u) } \, du=0`

As we already know that

`(dr(t))/(dt)=r^(')(t)`

Thus we can say that

`(dr(u))/(du)=r^(')(u)`

`dr(u)=r^(')(u)du`

Now,
`\int\limits^0_T {r^(')(u) } \, du =[r(u)]_(0)^T=r(t)-r(0)`

`r(t)-r(0)=0\\`

`r(t)=r(0)=(0,0)`

So at time T the bee will be at (0,0).