Madison J.

asked • 05/31/22

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 5.


y = x

y = 4

x = 0


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Doug C. answered • 05/31/22

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Patrick F. answered • 05/31/22

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Because of limitations with wyzant's formatting. I am putting the integral here, but you should read my comments below first. This is just the end of the problem:

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Patrick F.

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If you look at sketches of the functions you provided you will see that they make a triangle. When you rotate the triangle around the line y=5, you create a solid shape. To get the total volume you can imagine a series of discs, with the minimal width of dx, and then use integration to add up all these disks. The disks are shaped like very, very skinny donuts. The hole of the donut has a radius of 1 (the distance between y=4 and y= 5), and lets call it r1. The first problem is how to get the radius of the entire donut, lets call that r2 . Well, it is related to the line y=x. At this point I think it would be useful to shift the entire diagram down 5 units so that it rotates around the x axis. The volume will not change when you do this. Hopefully you can see that the radius of the entire disk, r2, is y = x - 5. You should produce this integral (see my next comment) : The working out of this should be trivial. I recommend you make a sketch, play around with it in your imagination. See how shifting can simplify the problem without changing the volume itself. Finally send your work to us here so we can take a look.
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05/31/22

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