Use the properties of integrals to verify the inequality

[MarkSA]

Hello,

Use the properties of integrals to verify the inequality without evaluating the integral:

pi/6 <= integral of pi/6 to pi/2 of: sinxdx <= pi/3

It looks like I would need to use one of the comparison properties of the integral, but I can't figure out how I would do so. My book does not give any sort of explanation for them. The one that looked promising was:
"If m <= f(x) <= M for a <= x <= b, then:
m(b - a) <= integral from a to b of f(x)dx <= M(b - a)"
I don't know what m and M are or really what the property means.

Any suggestions?

 

[royhaas]

Use the fact that 0<=sin(x)<=1 over the range you need.

 

[MarkSA]

Well,
0 <= sinx <=1 when 0 <= x <= pi
which includes [pi/6,pi/2]

I can show that the integral is >=0 by using one of the comparison properties. Not sure how to show that it has to be between pi/6 and pi/3 though?

Not sure where to go from here...

 

[MarkSA]

Nevermind. I figured this out.