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Numerade Educator

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Problem 64 Hard Difficulty

Use Property 8 to estimate the value of the integral.

$ \displaystyle \int^{2\pi}_{\pi} (x - 2\sin x) \,dx $

Answer

$\pi^{2} \leq \int_{\pi}^{2 \pi}(x-2 \sin x) d x \leq\left(\frac{5 \pi}{3}+\sqrt{3}\right)(\pi)$

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Video Transcript

the first thing we can do is take the derivative, which is one minus to co sign acts we know F prime a vaccine. Other words. That derivative is equivalent to zero when Coastline X is 1/2 which happens a pile five pi over three. Therefore test the value of off of pie, which we get as high after five pi over three, which we get to be approximately 6.97 you know, pies approximately 3.14 and then lastly, off of two pi the complete circle with just 6.28 approximately 3.14 times two, which gives our expression so five pied over three plus word of three times pyre through Portland for