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JH
Numerade Educator

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Problem 59 Hard Difficulty

Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct

$ \displaystyle \int_0^{2\pi} \cos^3 x dx $

Answer

$$
=\int_{0}^{0} 1-u^{2} d u=0
$$

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Video Transcript

for this problem would like to use a graph of the Inter grand to guess the value than a girl. And then we'LL actually a computer and a girl to see if it matches our guests. So let's go to a graphing calculator and see what the graph of the thean so grand, which is Cho sang kyu defects, Looks like for X between zero and pi over, too, till here's a graph, of course, Thank you. And so the integral will be the area underneath the curve from zero to pie. So over here, from zero to about one point five we see or which, in other words, pie, we see that we have positive area. Then, from about one point five to four point five, we see a larger negative area, and then afterward we see more positive area. So my guess is that this that an area underneath the curve down here is twice the area of each of these positive sections. So my guess is that the area's cancel out and I will get zero. So let's evaluate this integral to see what we actually get so we can rewrite this as in a girl's era to Pie Co sign Squared temps go sign and then using a potato grin identity. We can rewrite coastlines. Where is one minus science? Where? So let's go ahead and do that. So have one minus sine squared times co sign and this looks like we should go ahead and do a U substitution. So let's go ahead and do that take you to be signed so that do you is co sign. Also, our limits of integration will change. So the lower limit will become sinem zero, which is still zero, and the upper limit will become sign of two pi, which is zero to a swell. So after you substitution, we have the integral zero zero one minus you squared to you. And since the lower and upper limits of the integral of the same, they're both zero here. That means that the answer will be zero, which is consistent with the guests from the graph. So we have zero, and that's your final answer