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University of North Texas

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Problem 58 Medium Difficulty

Use the properties of integrals to verify the inequality without evaluating the integrals.

$ \displaystyle \frac{\pi}{12} \le \int^{\pi/3}_{\pi/6} \sin x \,dx \le \frac{\sqrt{3}\pi}{12} $

Answer

$\frac{1}{2} \leq \sin x \leq \frac{\sqrt{3}}{2} \frac{\pi}{12} \leq \int_{11}^{\pi / 6} \sin x d x \leq \frac{\pi \sqrt{3}}{12}$

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Video Transcript

So if we want to show this is true without actually integrating Cinemax, what we can do is the following. So let's assume so this is saying we're on the interval, so we're on this interval pie six too high Third. Now, if we were to kind of look at what sign is on this so this is kind of, you know, circle. So we have pie six here. Pi third here So you can see how sign increases on this interval because the Y values are getting larger, so that's going to tell us. So that implies that sign of pie six is going to be less than or equal to sign of X, which is going to be less than equal to sign of high third, at least on this interval Here. Now, over here. So sign up. I six would be one half sign up. I third is Route three over to now. What I'm gonna do is integrate all of this, so I just kind of do this in this way. So well, pie 62 pi third. So these constants that would just give us one half times so B minus a so pi third minus pi six less than equal to sign of the integral integral of sine of x dx by six to Peiser And then same thing over here. Route three over to, um either. Minus pi six. And now pie third minus Pi six is going to just be high six. Same thing over here by six. Yeah, and so that is going to give us pie 12 on the left side. We still have the integral here. And then over here, that would be route three over to route 3/12 pie. And so that is what we want to show.