84.a. Find the center of mass of a thin plate of constant dens...

Question

84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the center of mass of a thin plate of constant density covering the region between the curve. Y is equal to 1 divided by x, and the x axis from x is equal to 1 to x is equal to e to the power of 2. OK. So it appears for this particular problem, we're ultimately trying to solve for the center of mass of this thin plate that has constant density that covers the specific region between the curve, which is given to us for an expression for y is equal to 1 divided by x and the x axis from x is equal to 1 to x is equal to e to the power of 2. So now that we know that we're ultimately trying to solve for the center of mass for this thin plate. What we need to do first in order to solve for this problem is we need to focus our attention on the synthroid, and as we should recall, the synthroid of parentheses of X, Y F For a landmark between, so I'll just say between. Once again, the centroid parentheses of X, Y bar for a lambma between Y is equal to our function F of X. And the x axis. With a constant density, with a constant density on square brackets of A, B, will be given by the following expression or following equation, as we should recall, which states that X bar. is equal to M subscript Y. So once again, X is equal to M subscript Y divided by A. And that Y bar is equal to M subscript X divided by A. Fantastic. Where we should recall where the moment about the y axis is m subscript Y, which I'll call MY for short, is equal to the integral evaluated from A to B of x multiplied by F of XDX, and the moment about the x axis, which is mxm subscript x. is equal to 1/2 multiplied by the integral evaluated from A to B of parentheses of F of X to the power of 2. D X And the area equation, as we should recall, the area is equal to the integral evaluated from B to A of F of XDX. And we need to let, let's make a note that we need to let Y be equal to F of X be equal to 1 divided by X. Fantastic. So with that in mind, we now need to solve for the area. So, solving for the area, we will find that the area A is equal to the integral evaluated from 1 toe to the power of 2 of 1 divided by XDX, and this will be, once again, this will be equal to. The natural log of X. Which this will be evaluated from 1 toe to the power of 2. Which will be equal to the natural log of e to the power of 2 minus the natural log of 1, which will be equal to 2 multiplied by the natural log of E minus 0, which will be simply equal to 2. Fantastic. So, now, moving right along here, we now need to solve for the moment about the Y axis. So, we need to solve for MY and MY is equal to the interval evaluated from 1 toe to the 2 of X multiplied by 1 divided by X. D X, which will be equal to the integral evaluated from 1 to e to the power of 2. DX is equal to x. And this will be evaluated from, once again, 1 toe to the power of 2. Which will be equal to e to the 2 minus 1. So, therefore, we could state that X bar is equal to MY divided by A, which will be equal to e 2 minus 1 divided by 2. Fantastic. So, moving right along here, we now need to solve for the moment about the x axis. So we need to solve for mx and mx will be equal to 1/2 multiplied by the integral evaluated from 1 to e to the 2 of 1 divided by x in parentheses to the power of 2 dx. Which will be equal to 1/2 multiplied by the integral evaluated from 1 to e 2 of 1 divided by x 2. CX and this will be equal to 1 divided by 2 multiplied by the integral evaluated from 1 to e to the 2 of x to the power of -2. And this will be equal to drum roll, 1 divided by 2 multiplied by parentheses of x to the power of -1. Divided by -1. Evaluated from 1 toe to the power of 2. And this will be. Equal to drum roll 1 divided by 2 x. Evaluated from 1 to E to the power of 2. And this will be equal to -1 divided by 2 multiplied by e to the 2, because once again we're plugging in our variables for the evaluation, so it'll be equal to 1 divided by 2 multiplied by e to the 2 minus 1/2. Which will be equal to 1 divided by 2 or 1/2 minus 1 divided by 2 multiplied by e to the power 2, which will be equal to. Our final final expression in its most simplified form, which will be equal to e 2 minus 1 divided by 2 multiplied by e 2. So therefore we could safely write that y bar is equal to mx divided by a, which will be equal to e 2 minus 1 divided by 2 multiplied by e to the 2 divided by 2, which will be equal to and we simplify and solve, will be equal to e to the power of 2 minus 1. Divided by 4 multiplied by e to the power of 2. So therefore, our synthroid, and let's write that down, our synthroid. Is drum roll. So our final answer for the synthroid is parentheses of e to the 2 minus 1 divided by 2, e 2 minus 1 divided by 4 multiplied by e to the power of 2, and that's it. That is our final answer. Hooray, we did it. So going back up to the top to look at. The problem itself and to recap what we've learned, that will mean that once again, Our final answer without a doubt will be as follows. So once again, our final answer is parentheses of. E 2 minus 1 divided by 2, E to the 2 minus 1 divided by 4 multiplied by e to the 2. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.

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