How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #x=4y^2#, y = 1, x = 0 revolved about the yaxis?
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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 = 4y^2), (y = 1), and (x = 0) revolved about the yaxis:

First, sketch the region bounded by the curves (x = 4y^2), (y = 1), and (x = 0) to visualize the shape.

Identify the limits of integration for the cylindrical shells. In this case, the region is bounded by (y = 0) and (y = 1), so the limits of integration for y will be from 0 to 1.

Set up the integral for the volume using the formula for cylindrical shells: [V = \int_{a}^{b} 2\pi r h , dy] where:
 (a) and (b) are the limits of integration for (y),
 (r) is the distance from the axis of revolution to the shell (in this case, it's the value of (x)),
 (h) is the height of the shell (in this case, it's the change in (y) or (dy)).

Express (x) in terms of (y). Since (x = 4y^2), (x) can be expressed as (x = 4y^2).

Determine the height of the shell. In this case, the height is the change in (y), so (h = dy).

Determine the radius of the shell. The radius, (r), is the distance from the axis of revolution (the yaxis) to the shell, which is simply the value of (x).

Substitute the expressions for (r) and (h) into the volume integral and integrate with respect to (y) from the lower limit to the upper limit.

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.
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.
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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