How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=8-x^2#, #y=x^2# revolved about the x=2?

Answer 1

Please see below.

In the following picture, the region is shown in blue. The points of intersection of the two curves are #(+-2,4)#. The axis of rotation (the line #x=2#) is in red.

To set up for the method of shells, a slice has been taken parallel to the axis of rotation. The thin side (the thickness of the shell) is #dx#. The slice has black borders and the radius of rotation is shown as a dashed black line.

The Volume of a representative shell is

#2 pi r h "thickness"#

Since the thickness is
#dx#

(the variable we will work with is #x#) we note that #x# varies from #-2# to #2#.

The radius of the shell is
#r = 2-x#.

The height is the upper curve minus the lower curve

#h = (8-x^2) - (x^2) = 8-2x^2#

So the volume of the solid of revolution is

#V = int_-2^2 2piunderbrace((2-x))_r underbrace((8-2x^2))_h overbrace(dx)^("thickness")#

# = 2pi[128/3] = (256 pi)/3#

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Answer 2

To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by ( y = 8 - x^2 ) and ( y = x^2 ) revolved about the line ( x = 2 ), follow these steps:

  1. Determine the limits of integration by finding the points of intersection between the curves ( y = 8 - x^2 ) and ( y = x^2 ).
  2. Choose a representative vertical strip within the region of interest, parallel to the axis of rotation (in this case, the ( x )-axis), with thickness ( \Delta x ).
  3. Find the radius of the cylindrical shell, which is the distance from the axis of rotation (line ( x = 2 )) to the function.
  4. Determine the height of the cylindrical shell, which is the difference between the upper and lower functions.
  5. Express the volume of the cylindrical shell in terms of ( x ), then integrate this expression from the lower limit of integration to the upper limit.
  6. Multiply the result by ( 2\pi ) to account for the rotation around the axis of rotation.

This procedure yields the volume of the solid obtained by rotating the region bounded by ( y = 8 - x^2 ) and ( y = x^2 ) about the line ( x = 2 ).

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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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