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For the following exercises, use a calculator to draw the region, then compute the center of mass $(\overline{x}, \overline{y}) .$ Use symmetry to help locate the center of mass whenever possible.The region between $y=2 x^{2}, \quad y=0, \quad x=0$ and $x=1$

$\left(\frac{3}{4}, \frac{3}{5}\right)$

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 6

Moments and Centers of Mass

University of Michigan - Ann Arbor

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Okay, so we have wise equal to two X and two X squared Y is equal to zero. And let's see, explica todo to one. Okay. Were asked to draw out the region and in computer center of Mess, and we could use somebody to help locate the region. Okay, so let's draw out our graph first. Well, two X squared is a problem. We have something like this, and then it cuts up between Joe and one. So it's returned. One is over here. So we're looking at this region over here. Okay, Well, looking at this, it seems like we don't have any cemetery, so because we don't have any cemetery we had to use, um, any properties or any shortcuts of calculating it, So we're gonna do it directly. So we're gonna take export to equal interval from Gerald to one of X times. Why, that's two x cubed over the integral of just Joe's one from of exports eggs. The X does that Give me that. Gives me. Mmm mmm. Oh, it is it to over here to actually Cassidy Stew's. Okay, So I had just the interval from Joe's one of Excuse that you explore over four over X three over three, evaluated at one, and it's usually one over four over one over three, which is equal to three over four. Now let's find White Boy. It's equal to 1/2 of the interval, just one of two X squared times X ray just to be ex. All right, which is execute the X over in ago. Joe's one of two X squared DX. What does that give me? That gives way. Oh, wait, no, I'm squaring it right, I'm staked. So I get actually to exploit the power of Jews. That's four x for X over Indigo from Joe's one. Just why would just two x squared x Okay, so we get well, our 1/2 can't so that this because it too. And then I choose chances that would get X over five over five extra para fireball in five over X to part three over three about looting at one and tell excuse me. Three over five So our center mass becomes really over four comma, three over five

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