# How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #x=sqrt(y)#, x=0, y=4 about the x-axis?

(0, 0) and (2, 4).

The shell has a paraboloid-hole in the middle. The base is circular,

with radius ( range of y ) 4 units and the height is ( range of x, from 0

to 2 ) 2 units.

The hole volume has to be subtracted from the volume

of a right circular cylinder, of radius 4 and height 2 units..

the limits, from x = 0 to x = 2

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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 (x = \sqrt{y}), (x = 0), (y = 4) about the x-axis, we integrate the volume of each cylindrical shell along the x-axis. The formula for the volume of a cylindrical shell is (V = 2\pi x h \Delta x), where (x) is the distance from the axis of rotation to the shell, (h) is the height of the shell, and (\Delta x) is the width of the shell along the x-axis.

We express (x = \sqrt{y}) as (y = x^2) and (y = 4) as (x = 2). The radius of each cylindrical shell is (x) and the height is (4 - x^2). We integrate from (x = 0) to (x = 2):

[V = \int_{0}^{2} 2\pi x(4 - x^2) , dx]

[= 2\pi \int_{0}^{2} (4x - x^3) , dx]

[= 2\pi \left[\frac{4x^2}{2} - \frac{x^4}{4}\right]_{0}^{2}]

[= 2\pi \left[2x^2 - \frac{x^4}{4}\right]_{0}^{2}]

[= 2\pi \left[2(2)^2 - \frac{(2)^4}{4} - 0\right]]

[= 2\pi \left[8 - 4\right]]

[= 8\pi]

So, the volume of the solid obtained by rotating the region about the x-axis using the method of cylindrical shells is (8\pi).

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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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