Disk or Washer method: y = x, y = 5x, y = 20 about y-axis

nbg273

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Jan 27, 2017
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Hi guys,


I'm having a little trouble setting this problem up:


"Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the​ y-axis. y = x, y = 5x, y = 20"


I can solve for the volume once I have the integral set up, but I'm a little confused on how to get the boundaries, as well as which function is being subtracted from the other.

Normally I'll get two "y = ..." and two "x = ...", but I have three "y = ..." this time.


Any tips would be great!


Thank you.
 
I'm having a little trouble setting this problem up:

"Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the​ y-axis. y = x, y = 5x, y = 20"
What have you done, so far, thinking about setting up the integral(s)?

Have you graphed the region to be rotated, on the xy-plane? It's in Quadrant I. This helps to visualize the shape of the object. (It's an upside down cone, with a smaller, cone-shaped hollow space in its center.) Knowing the general shape, you can think about what the individual pieces look like.

For these pieces, you have a choice between using washers or shells.

With washers, you would integrate with respect to dy, and that would require solving y=5x for x because that x is the inner radius of each washer.

With shells, you can use the functions as given, integrating with respect to dx, but you'll need to sum a pair of integrals.

Have you already decided on a method? In your subject line, you wrote "Disk or Washer", but those are two names for the same method (washers are disks, with a hole in the center, and integration stacks them, to form the solid). The other method uses shells (sort of like wrapping outer tubes around inner tubes, to build up the solid.)

Either way (washers or shells), draw a representative slice, on your graph. That is, draw a sample of one rectangular region that will be rotated around the y-axis. You can then use the diagram as a guide for determining what gets subtracted from what and where the boundaries lie.

If you're unable to graph the given equations, to see the region being rotated, or you're not sure how to draw one representative "slice" whose rotation forms a single shell (or washer), let us know.
 
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Were you able to graph the given information? Did you figure out the dimensions of the representative rectangle (slice to be rotated)? :cool:
 
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