Moment about y-axis:
A thin plate of constant density δ ...

Question

Moment about y-axis:

A thin plate of constant density δ = 1 occupies the region enclosed by the curve

y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

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. A thin sheet of uniform density row is equal to one occupies the region in the first quadrant bounded by the curve. Y is equal to 50 divided by 4 x plus 5. The vertical line x is equal to 4, and the coordinate axes. Compute the moment of this sheet about the y axis. OK, so it appears for this particular prom, we're ultimately asked to compute the moment of the sheet about the y axis based on all the information that is provided to us by the prom itself. So Now that we know what we're ultimately trying to solve for. Our first step that we need to take is we need to note based on the information that is provided to us by the prom itself, we can safely say that the sheet occupies 0 is less than or equal to x and x is equal to 4.0, which is less than or equal toy, which is less than or equal to. 50 divided by 4 X plus 5, and it has a density, so it's with a density, row is equal to 1. So, the moment about the y axis formula will be, as we should recall. So once again, we need to recall and use the formula for the moment about the y axis, which states that M subscript Y, which I'll simply call MY, MY is equal to double integral. Of X multiplied by row multiplied by DA. Which will be equal to the integral evaluated from x is equal to 0 to 4 of the integral evaluated from Y is equal to 02 50 divided by 4 x plus 5. Of drum roll. X multiplied by DY multiplied by DX, which will be simply equal to, once again, it's just going to be simply equal to the integral evaluated from 0 to 4 of 50 x divided by 4 X plus 5 DX. Fantastic. We are on a roll here. So now we need to compute the interval, so we need to recall and use our substitution method, and when we do, we will state that U for this case will be equal to 4 x plus 5, which will mean that DU will be, so once again, DU will be equal to. 4 d x. So that will mean therefore that and x will equal to u minus 5 divided by 4. And once again this is using our substitution method technique. So we need to note that when x is equal to 0, that will mean that u is equal to 5. And when X is equal to 4, that will mean that U is equal to 4 multiplied by 4 plus 5, which is equal to 21. So therefore we could safely say then that the integral evaluated from 0 to 4 of 50 divided by 4 x plus 5 DX will be equal to the integral evaluated from U is equal to 5 to 21 of 50 multiplied by parentheses of u minus 5. Divided by 4 divided by u multiplied by duu divided by 4, which will be equal to 25 divided by 8 multiplied by the integral evaluated from 5 to 21 parentheses of 1 minus 5 divided by UDU. Fantastic. Thus, we can go ahead and write that MY will be equal to 25 divided by 8 multiplied by parentheses of u minus 5 multiplied by the natural log of the absolute value of u evaluated from 5 to 21. Which will be. Equal to 25 divided by 8 multiplied by parentheses of parentheses of 21 minus 5 multiplied by the natural log of the absolute value of 21 minus parentheses of 5 minus 5 multiplied by the natural log of the absolute value of 5. Which will be, once again, you have to keep going, which will be equal to when we write our final answer in its most simplified form, it will be simply equal to 25 divided by 8 multiplied by parentheses of 16 minus 5 multiplied by parentheses of the natural log of 21 minus the natural log. Of 5, and we can get rid of the absolute value signs because anything inside the absolute value sign will become a positive value. So With that said, we now need to simplify even further, so I lied temporarily, because we could simplify this even more. So we could take 25 divided by 8, multiplied by parentheses of 16 minus 5 multiplied by parentheses of the natural log of 21 minus the natural log of 5. And this will be equal to. 50 Minus 125 divided by 8 multiplied by the natural log of 21 divided by 5 and that's it that's our final answer. Somy is equal to 50 minus 125 divided by 8 multiplied by. Drum roll, the natural log of 21 divided by 5, and that's it. That is our final answer. Hooray, we did it. So going back to look at the prom itself, we could safely say once again that our final answer without a doubt will be. 50, so it's gonna be 50 minus 125 divided by 8, multiplied by the natural log of 21 divided by 5, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Moment about y-axis:A thin plate of constant density δ = 1 occupies the region enclosed by the curvey = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

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Moment about y-axis:
A thin plate of constant density δ = 1 occupies the region enclosed by the curve
y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.