How do you know when to use the shell method or the disk method?

Answer 1

You'll get the same answer either way.

Sometimes one leads to an integral that a particular person finds easier to evaluate, but what is easier varies between people.

I have a preference for doing a single integral.

So if I have to find the volume of the solid generated by revolving the region bounded by #x=0#, #y=x^2#, and #y=-x+2# around the #y#-axis, I would use shells because there would only be one integral to evaluate. (Disks would require two: one from #y=0# to #y=1# and another from #y=1# to #y=2#.)
Taking #y=0#, #y=x^2#, and #y=-x+2# around the #x#-axis, I would use shells to avoid doing two integrals even though it would require me to rewrite the curves as functions of #y#.

Again, that is my preference. As a student, perhaps because we learned disks/washers first, I preferred them.

In the end it doesn't matter. If you set it up one way and don't care for the looks of the integral, try setting it up the other way.

It is possible that you need to use a function whose inverse you can't find. For instance if one boundary of the region is #y=x^4-8x^3+x+22# we do not want to try to rewrite this as a function of #y#. So we'll use the set-up that keeps things in terms of #x#.
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Answer 2

You can determine whether to use the shell method or the disk method to find the volume of a solid of revolution by considering the axis of rotation and the orientation of the slices.

  1. Shell Method:

    • Use when the axis of rotation is horizontal or vertical (parallel to the slices).
    • Slice the region perpendicular to the axis of rotation.
    • Each slice is a cylindrical shell (a hollow cylinder).
    • Integrate along the axis of rotation (usually (x) or (y)).
  2. Disk Method:

    • Use when the axis of rotation is perpendicular to the slices.
    • Slice the region parallel to the axis of rotation.
    • Each slice is a disk (or washer if there's a hole).
    • Integrate perpendicular to the axis of rotation (usually (x) or (y)).

To decide which method to use:

  • Visualize the region and the axis of rotation.
  • Determine the orientation of the slices needed for each method.
  • Choose the method that aligns with the orientation of the slices and the axis of rotation.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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