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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.$[{T}]$ Region between $y=\sqrt{x}, \quad y=\ln (x), \quad x=1$ and $x=4$

$(2.38,1.15)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 2

Applications of Integration

Section 6

Moments and Centers of Mass

Integrals

Integration

Missouri State University

Harvey Mudd College

Baylor University

Boston College

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in the following problem, we went to find the center of mass of the regions in between the grafts off bicycle, just quarter defects and wise people to liberate natural logarithms off Alex for ex imaging white wanting for. So let's first start off by making a sketch of the graph. So we know that why sickle to excess square looks something like this unless actually pinpoint the point one coma want. No, we don't. We know that l natural algorithm effects is zero with one and that it continues to grow. So we're here. We have the 0.0 Sorry. One comma, sirrah. And over here we have the intersection at zero comma zero. So we're looking at this point right here. Now, I'm just establish that this is a point x equal to fort, so we want to refine center of mass of this region right here. Now we have the German that we need to find this interference with its white with x X and y coordinate. And we know that it's X coordinate. It's gonna be equal to the moment. With respect to the XX is sorry to the Y axis divided by the total must've the system and why it's gonna be equal to the moment would respect to the X axis divided by the total mass of the system. So let's first thing the total mass of the system and which is equal to the integral from a TV of the density of the quake off the function right here Times If X minus G affects, Yanks in our case will have em equal to the into girl from 1 to 4 off row. The density trumps a square with defects minus ln effects. Yes, the first integral is easy to compute, so we'll have row times. The integral over square root effects is 2/3 temps extra three halfs if we use Powerball and now the integral of negative natural logarithms effects by using a graphing calculator or computer software program to find it is made of X times. Alan Effects was X. None of this is being evaluated from 1 to 4. If the approximate values will end up with them equal to 2.1 to 1 row. Now I'm going to use another whiteboard. You calculate the moment with respect to the white necks this physical two days to the Inter integral from a TV of rope times X temps ffx minus G affects the ex. Now let's have sifted. The balance overhears We'll have the integral from 1 to 4 of row temps X Times Square with defects minus Ellen effects The ex The moment would respect to the Y axis can be written as the integral from 1 to 4 of Roe Times X to the three cafs minus X temps. Ellen effects the ex The first, integral again can be computed easily. So this is gonna be equal to two over five terms. Ex rest of the 5th 5 over too. And again, we need to use a graphing calculator computer somewhere program. Do you actually find out this integral which is equal to negative one over true excess where Ellen Effects plus one over four excess where all of this is being evaluated from 1 to 4. If we substitute, these values will end up with the moment with respect to the Y axis equal to 5.6 times rope. Now let's find the moment would respect to the X axis? This quantity is equal to 1/2 attempts to integral from from a TV of rope times F f X square minus gene affects the square Yanks. If we substitute, we'll have Dane tickle from 1 to 4 of road times the square of this world. The fix is just X and the square natural logarithms just ellen off excess Where the ex now this integral the 1st 1 just becomes square. It affects Sorry, excess worried, Divided by two. The second, Integral will again need a use of a graphing calculator, a computer somewhere to find it. And it is equal to X terms. Ellen affects a square plus two X temps Ellen effects minus do X, and this is being evaluated from 1 to 4. And if you plug in all of its values, you'll find out that the moment would respect to the X axis is equal to 2.452 Temps Road. Now that we have determined this Valley's, we can actually start plugging in to find the coordinates for the moment, for the center from s. Sorry. So the 1st 20 is gonna be 5.6 Thames row derided by 2.1 truth. Sorry to 0.1. Shoot one rope coma, 2.452 temps around, which is just this quantity over here, divided by 2.1 to 1 roll. So the coordinates the excrement of disinterred messages. 2.3 years. 86 after simple fighting, this fraction of mercury and the other one is 1.156 And this one right here is a solution to this question.

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