How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y^2=8x# and x=2 revolved about the x=4?
Please see below.
Here is a sketch of the region. To use shells, we'll take a representative slice parallel to the axis of rotation. (Parallel to the line
The slice is taken at some value of
The volume of the representative shell is
In this case,
the radius
The height of the slice is the upper
Solving
So,
The volume of the representative shell is
# = 8pisqrt2 int_0^2 (4x)sqrtx \ dx#
# = 896/15 pi#
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To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by (y^2 = 8x) and (x = 2) revolved about the (x = 4) axis, follow these steps:

Identify the axis of rotation and the region being rotated.

Determine the limits of integration for the cylindrical shells. In this case, the region is between (x = 2) and the yaxis, so the limits of integration for (x) will be from 0 to 2.

Set up the integral for the volume using the formula for the volume of a cylindrical shell:
[V = \int_{a}^{b} 2\pi \cdot r \cdot h ,dx]
where (r) is the distance from the axis of rotation to the shell, and (h) is the height of the shell.

Express (r) and (h) in terms of (x). Since the axis of rotation is at (x = 4), the radius (r) is (4  x). The height (h) can be expressed as (y), which is the distance between the curves (y^2 = 8x) and (x = 4), so (h = 2\sqrt{2x}).

Substitute (r) and (h) into the volume formula and integrate with respect to (x) from 0 to 2:
[V = \int_{0}^{2} 2\pi \cdot (4  x) \cdot 2\sqrt{2x} ,dx]
 Evaluate the integral to find the volume of the solid.
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When evaluating a onesided 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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