How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=3x^4#, y=0, x=2 revolved about the x=4?

Answer 1

#(128/15)pi# cubic units

The volume of this annular solid, open at y = 0 and y = 48 and

appearing as an inverted funnel is

#pi int(4^2-(x-4)^2) d y#, from y=0 to y = 48.
#=pi int (2 x - x^2) d y#, from y=0 to y = 48..
#=pi int (2 (y/3)^(1/4) - (y/3)^(1/2) )d y#, from y=0 to y = 48..
#=pi [2(y/3)^(5/4)/(5/4) - (y/3)^(3/2)/(3/2)]#. between y = 0 and y = 48
#=.pi [2(48/3)^(5/4)/(5/4) - (48/3)^(3/2)/(3/2)]#
#=.pi [256/5 - 128/3)]#
#=(128/15)pi# cubic units.
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Answer 2

To use the method of cylindrical shells, we consider thin vertical strips parallel to the axis of revolution. The height of each shell is the difference between the outer radius and the inner radius, and the thickness is the differential change in x.

The outer radius is the distance from the axis of revolution (x = 4) to the curve farthest away (y = 3x^4). Thus, the outer radius is 4 - x.

The inner radius is the distance from the axis of revolution to the curve closest to it, which is the x-axis. Therefore, the inner radius is 4 - 0 = 4.

The height of each shell is the difference between the outer radius and the inner radius, which is (4 - x) - 4 = 4 - x.

The differential thickness in the x-direction is dx.

The volume of each cylindrical shell is given by 2π(radius)(height)(thickness). Therefore, the volume element is 2π(4 - x)(4 - x^4)dx.

To find the total volume, integrate the volume element over the interval where the curves intersect, which is from x = 0 to x = 2:

∫[0 to 2] 2π(4 - x)(4 - x^4) dx

Evaluate this integral to find the total volume of the solid obtained by rotating the region bounded by y = 3x^4, y = 0, and x = 2 about the line x = 4.

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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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