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 ∫(0-T) r'(u)du = 0? Also, what does the quantity ∫(0-T) ||r'(u)||du represent?

Answer 1

Please help

Answer 2

what does (0-T) represent?

Answer 3

integral from 0 to T

Answer 4

sorry it's the only way I know how to type it

Answer 5

so those are the limits, ok

Answer 6

do you recall the definition of: \[\int_{a}^{b}f(x)~dx=??\]

Answer 7

the fundamental thrm of calculus is also what it is called

Answer 8

It's F(b) - F(a)

Answer 9

correct, so if F(b)-F(a) = 0, then either a=b, or F has the same value at multiple input levels similar to a periodic function.

Answer 10

for example, cos(pi) - cos(0) = 0 since cos(pi)=cos(0)

Answer 11

but the question is, what does taking the integral of a function tell us?

Answer 12

it can tell you what the area under a curve is

Answer 13

and my example is spose to read cos(2pi) = cos(0), but the site is lagging for me

Answer 14

it's ok

Answer 15

the area under a curve, yes ... but that reflects displacement

Answer 16

ever wonder why in some cases we get a negative area? and in other cases we dont?

Answer 17

Yes because the integral of velocity is displacement

Answer 18

right?

Answer 19

negative displacement ?

Answer 20

when we want to find the area between 2 curves, area is a positive value always ... its an absolute value. And in those cases we have to concern ourselves with the places that one curve crosses over the other.

Answer 21

displacement on the other hand reflects is a vector, it has magnitude and direction.

Answer 22

F(b)-F(a) = 0 assumes that at a and b, we are in the same place. Either a = b, or we have simply ended were we started.

Answer 23

Then the location of the bee is still the origin at time T?

Answer 24

yes, at time T, the bee is simply back in the same place it was at when the time was equal to 0

Answer 25

But why substitute u in for t for the integration? That's the most confusing part. Does that have any effect on the result?

Answer 26

0 and T are values that the variable u can take on. the function itself is dependant on the value of u ...

Answer 27

What is the difference between
∫(0-T) r'(u)du = 0

and

∫(0-T) r'(t)dt = 0

Why did they substitute ?

Answer 28

hmm, im not absolutely sure, but i think it has something to do with what is called a 'dummy' variable - meant to avoid confusion between the overuse of the letter T

Answer 29

oh ok .. well thank you !

Answer 30

good luck