Let R be the region in the first quadrant enclosed by the hyperbola #x^2 -y^2= 9#, the x-axis , the line x=5, how do you find the volume of the solid generated by revolving R about the x-axis?

Answer 1

#44/3 pi#

A small vertical strip width #Delta x# within R will have revolved volume as follows

#Delta V = pi y^2 Delta x#

So
#V = pi \int_3^5 y^2 \ dx#

#= pi \int_3^5 x^2 - 9 \ dx#

#= pi [ (x^3)/3 - 9x ]_3^5 #

#= pi [{ (5^3)/3 - 9(5)} - {(3^3)/3 - 9(3)} ]#

#= 44/3 pi#

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Answer 2

To find the volume of the solid generated by revolving the region ( R ) about the x-axis, we can use the method of cylindrical shells.

  1. First, sketch the region ( R ) enclosed by the hyperbola ( x^2 - y^2 = 9 ), the x-axis, and the line ( x = 5 ) in the first quadrant.

  2. Determine the limits of integration for the x-values. From the equation of the hyperbola ( x^2 - y^2 = 9 ), we find the x-intercepts which are ( x = \pm 3 ). Since we are interested in the first quadrant, the limits of integration for x-values are from 0 to 3.

  3. For each x-value in the interval [0, 3], determine the corresponding y-values on the hyperbola using the equation ( x^2 - y^2 = 9 ), which gives ( y = \sqrt{x^2 - 9} ).

  4. Now, the volume ( V ) generated by revolving ( R ) about the x-axis can be found using the formula for the volume of a solid of revolution:

[ V = \int_{0}^{3} 2\pi x \cdot y , dx ]

  1. Substitute the expression for ( y ) in terms of ( x ):

[ V = \int_{0}^{3} 2\pi x \cdot \sqrt{x^2 - 9} , dx ]

  1. Evaluate the integral from 0 to 3 to find the volume.

[ V = \left[ \frac{2}{3}\pi (x^2 - 9)^{\frac{3}{2}} \right]_{0}^{3} ]

[ = \frac{2}{3}\pi ((3^2 - 9)^{\frac{3}{2}} - (0^2 - 9)^{\frac{3}{2}}) ]

[ = \frac{2}{3}\pi ((9 - 9)^{\frac{3}{2}} - (-9)^{\frac{3}{2}}) ]

[ = \frac{2}{3}\pi (0 - (-9)^{\frac{3}{2}}) ]

[ = \frac{2}{3}\pi \cdot 9 ]

[ = 6\pi ]

So, the volume of the solid generated by revolving region ( R ) about the x-axis is ( 6\pi ) cubic units.

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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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