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Problem 37 Hard Difficulty

Find the centroid of the region bounded by the curves $ y = x^3 - x $ and $ y = x^2 - 1 $. Sketch the region and plot the centroid to see if your answer is reasonable.

Answer

$=(-0,2,-0.34)$

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AS

Atlas S.

March 10, 2021

The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system.

Video Transcript

in this problem Where us To find the central region bounded by Dacres Excuse minus X and X. Great grindstone. And that will be be shaded region. Let's start by finding the area that region. Basically, you're forming 10 strips, what with DX and know where someone goes up. So we're gonna integrate those areas. As you can see, the limp so far, X r expert Negative War two ex prostitute one. So the limits of the integral will be negative. Alert one. Um, Thea per function here is excuse minus X. So that would be our FX and excreta. Minus one is lower function. So that revealed your pecs of we have for area intergalactic if x minus two effects. And that would be excused. Mind sex when sex card plus warm DX, we can easily find the entire day with hope that one s excursion over four months extra to fort over for one sec. Skirted two minus x cube over three plus X or exchanges between Negative on it one. And from this we find the area to be four or three. You know that X bar will be off to form one or area in trouble for me to be or AIDS negative one on these positive or X times after effects ones G affects the X. We have everything that we need. So let's display goes in. X bar will be one or area where areas for over three. So what over that would be three over four. Interval from mega to 12 on X times. FX is execute minus X and you're fixed. They scream once once, so we have extra to fort minus X cried minus X cube plus X X entire It'll that would be three or four exit 50 or five minus X tube over three minus extra to ford over four plus X card or two. Where exchange between negatory on a war when we applied in 80 to 1 and one And do the calculation refund export to be negative. Warn over five. That is negative 50.2. All right, You know what the Y bar will be up to? Form water or area interning for me to be one house off if scared off. X minus G skirt effects DX. So people like what we haven't. We have three or four as one of our area Internal from negative. Want one? What have off X cube mind specs squared minus X grade. My swan scream The X We can just take this 1/2 outside each ago, since that's just a constant. You don't have white bar as three or eight inter. Go from negative 1 to 1 extra. 86 minus three extra Ford plus two years squared minus, uh one. The eggs that is equal to three or eight x 37 over seven minus three over five, actually, five plus X cube minds ex were exchanges between negative one and one. Why bar from this is then three or eight times to over 76 over five. Be fine wine bar to be negative 12 or 35 that is approximately equal to negative 350.34. So from a species as center of mass off that region is that at negative point to a negative point. 30 four