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Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.$y = \sin x$ , $y = 0$ , $0 \le x \le \pi$

$(\overline{x}, \overline{y})=\left(\frac{\pi}{2}, \frac{\pi}{8}\right)$

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Heather Z.

Oregon State University

Michael J.

Idaho State University

Boston College

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Video Transcript

in this problem were as to find the central mass off the region bounded by terrorists. Wise ical sine X y sickle zero and X equals zero in X equals High said I ever be shaded region and usually we can see that the central mess will be so work around here. Now, let's calculate it analytically. We first need to calculate the area that region. The idea is that we are forming 10 strips with Whit off D X. And now we're something those up meaning that we're gonna integrated. And as you can see, exchanges from zero to plants of the limbs off or internal will be between zero to pie. And we're gonna some those 10 ships up. So we had an sign Eggs times D X, All right entirely over to off sign eggs Is people to negative. Of course, I next. So we have negative co sonics and exchanges between is your apply. From this? We see that then the area is equal to two Now let's cock glad export. We not an export is one over area internal from a to B x, ffx the eggs and playing everything in. We didn't see that expert will be cooked to one house of you struggle from 0 to 2 pi We have ex scientifics d X in urges Hold this. We're going to use interrogations. What parts were going to say that access sickle to you So d X would be equal to you and sign x d x physical to Devi sort of negative of Coastline X would be equal to me so that we could write explore as one How off? Negative Exco Sonics Minus in Tegel Minus sign Sorry Co sign X d x where exchanges between you're implying we confer to write this one as one house off So the Knicks it comes from here Uh, minus x co sign X and exchanges between urine pie. Calculating this plugging in pine plugging in zero BC Not export is located at high over too. So somewhere right along this line. All right, now let's talk about why bar you know that why warrants he could to one or a intern go from a to B one. Health off to function Squared the eggs That is the Nico to one health off Since when I was just a ghost. Unless they get outside the integral what have in Tegel From zero to pie We have effects of sine x or we have sine squared x t x We can write this one s when we were four times integral from zero to pie one house one minus co Sign of two x d x that it's one minus her side. All right now we can easily take the integral One else is just a constant again So let's take it outside. Intercollegiate, have wife are as what? Eight Integral from zero to pie Ah, one once costar next actual list Get rid Oft interred Gilded right This one is thanks minus sign to x over to where it changed to easier on pie So why bar is an equal to 18 off ply minus Sign to pine over too sign to 0.0. So that zero So we can see that Why bar is that pile of eight sort of central mess off the system will be at pie or two and pie 08

Heather Z.

Oregon State University

Michael J.

Idaho State University

Boston College

Lectures

Join Bootcamp