Proving Integral Homework Statement: x>0

[ptolema]

Homework Statement

prove that

for all x>0

Homework Equations

-1 $$\leq$$ sin t $$\leq$$ 1

The Attempt at a Solution

the area under the graph is increasing as x increases
also, i tried to write it the sigma way:

then take the limit as n-->infinity
i got stuck trying to figure out how to work with sine in sigma notation, but I'm not even sure if my attempt would get anywhere

can anyone give me any pointers on how to do this?

 

[Whitishcube]

All you have to do is basically prove that the integrand is positive; since we know that the area under a positive function is positive, all you basically need to show is that the integrand is always positive. If it is not, then you need to prove that the positive area is greater than the negative area. A good place to start would be to graph the integrand.

 

[ptolema]

All you have to do is basically prove that the integrand is positive; since we know that the area under a positive function is positive, all you basically need to show is that the integrand is always positive. If it is not, then you need to prove that the positive area is greater than the negative area. A good place to start would be to graph the integrand.

i can't say that the integrand is always positive, and there are infinitely many values of t where sin t is negative. i know that the integrand approaches 0 as t-->infinity, but that may or not be important. is the boundedness of the integrand important?