Problem

For the following exercises, use a calculator to …

Problem 274 Hard Difficulty

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 $x=1$

Answer

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

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus

Chapter 2

Applications of Integration

Section 6

Moments and Centers of Mass

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Video Transcript

in the following problem. We want to find the center of mass of the region in between the functions. Why is equal to two X square and waste equal to zero for X in between zero and one. So let's first draw sketch for this graph. So over here we're going to draw the functional bicycle to Texas Square, which we know is the problem and the journal for myself. Gravel. It's gonna be this one right here. And let's just a point to substitute X just to pinpoint. So we'll have the point X is equal to zero in the point X equal to one. So let's find what is the white co ordinate for this point. So the Y sickles zero. We substitute that over here. Well, find when zero. I'm gonna use Rhett to determine at this point. And now at X equal to one y sickle to two. So we're here with a 0.2. Call me one. Sorry. One Coma too. The 0.0 commence here. Now the line y sickle to zero is just the X axis. Is this one right here? So the region aren't that we're trying to find its center of mess is this one right here? So we have drawn the function in the graph. Let's find out the center of mess. So we know that the center of mass it's located. But this coordinates and we know that this special cases of X and Y are located at the moment with respect to the Y Axis awaited. But the total mass of the system and the way coordinate it's located at the moment would respect to the ex access be raided by the mass of the system? So let's now calculate these three quantities we're here. So the start of by calculating the moment would respect to the wax. This is equal to the integral from A to B of the density of the density of the function. Tim's X temps, the upper function Buying is lower function times DX, in this case are bounce. We're gonna being from Syria to one Temps road Temps X. So FX is gonna be to excess square And gee, if X is gonna be zero, So we're just gonna leave that a serious now we have the integral from Syria to one of Roe terms to excuse the ex we can now perform this integral, which is released to compute, which is just gonna be extra fourth divided by four which in this case of was ballots are invalid from Syria to one. And if we succeed that these will have to roam times one over four, which is just roll over too. In their white words went to calculate the mass of the system. Just gonna be the integral from a TV of room tempts the upper function minus the lower one. The bounce are going to stay the same So from Syria to one times, Rowe says We know that ji, if X is equal to zero, would just substitute over here to exit square the ex. We can now perform this integral which is just X cube divided by three invalid from Syria to one. So the mass of the system is just your row terms one be better by three, which is just 2/3 of Rome. Finally, let's calculate the moment with respect to the X axis. Now this is equal to 1/2 times the integral from a TV of roe Times Square fffx minus the square of GI effects the ex We can now sit sit through their bounce. And since we know that GI effects zero, we'll just have to excess square square. This is equal to 1/2 Tim's room times for extra the fourth. Yes, sorry. Forgotten DX over here. So the moment would respect to the X axis. It's not just went too weak to Thames Row times this integral right here, which is just X to the fifth power divided by five validated from Syria to one which is just to row. He read it by five. Now we have calculated all of these three quantities. We can find out the location of the center of mess, which is going to be We know that X according it's gonna be at the moment of inertia. Would respect to the Y axis derided by M and why is gonna be the moment of inertia with respect to the X axis? You bet it by m. So the first term is gonna be a road divided by two divided by to Rome, divided by three the white coordinate It's going to be to row invited by five divided by two row, divided by three. The center of mass is just then we'll have three of Rome divided by four row coma. You wrote temps three debated by five, divided by to row. So in this case, the X coordinate is gonna be three divided by four. The white coordinates gonna be three divided by five. And this is a solution to this.

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