# How do you compute the volume of the solid formed by revolving the fourth quadrant region bounded by #y = x^2 - 1# , y = 0, and x = 0 about the line y = 4?

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To compute the volume of the solid formed by revolving the fourth quadrant region bounded by (y = x^2 - 1), (y = 0), and (x = 0) about the line (y = 4), you can use the method of cylindrical shells.

First, find the limits of integration by setting the equations equal to each other:

(x^2 - 1 = 0)

(x^2 = 1)

(x = \pm 1)

So, the limits of integration are from (x = 0) to (x = 1).

The radius of the cylindrical shell at any (x) value is the distance from the line of revolution (y = 4) to the curve (y = x^2 - 1), which is (4 - (x^2 - 1) = 5 - x^2).

The height of the cylindrical shell is the difference between the two functions, (y = 4 - (x^2 - 1) = 5 - x^2) and (y = 0), which is simply (5 - x^2).

So, the volume (V) of the solid can be computed using the formula for the volume of a cylindrical shell:

[ V = \int_{0}^{1} 2\pi r h ,dx ]

[ V = \int_{0}^{1} 2\pi (5 - x^2)(5 - x^2) ,dx ]

[ V = \int_{0}^{1} 2\pi (25 - 10x^2 + x^4) ,dx ]

[ V = 2\pi \int_{0}^{1} (25 - 10x^2 + x^4) ,dx ]

[ V = 2\pi \left[ \frac{25x}{1} - \frac{10x^3}{3} + \frac{x^5}{5} \right]_{0}^{1} ]

[ V = 2\pi \left( \frac{25}{1} - \frac{10}{3} + \frac{1}{5} \right) ]

[ V = 2\pi \left( 25 - \frac{30}{15} + \frac{3}{15} \right) ]

[ V = 2\pi \left( 25 - \frac{27}{15} \right) ]

[ V = 2\pi \left( \frac{375 - 27}{15} \right) ]

[ V = 2\pi \left( \frac{348}{15} \right) ]

[ V = \frac{696\pi}{15} ]

[ V = \frac{232\pi}{5} ]

So, the volume of the solid is ( \frac{232\pi}{5} ).

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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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