Home > Calculus > Determining the Volume of a Solid of Revolution

How do you find the volume of #y=3/(x+1)#, #y=0#; #x=0#; #x= 8# rotated around the x-axis?

Answer 1

Charles Arocha

#8 pi#

The region to be rotated around x axis is shown, shaded blue, in the picture shown below

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Answer 2

Emma Aldridge

To find the volume of the solid generated by rotating the region bounded by the curves ( y = \frac{3}{x+1} ), ( y = 0 ), ( x = 0 ), and ( x = 8 ) about the x-axis, you can use the method of cylindrical shells. The formula for the volume of a solid generated by revolving the region bounded by ( f(x) ), the x-axis, ( x = a ), and ( x = b ) around the x-axis is:

[ V = 2\pi \int_{a}^{b} x \cdot f(x) , dx ]

Substitute the given function ( f(x) = \frac{3}{x+1} ) and the limits of integration ( a = 0 ) and ( b = 8 ) into the formula and compute the integral.

[ V = 2\pi \int_{0}^{8} x \cdot \frac{3}{x+1} , dx ]

After integrating, you should get the volume of the solid.

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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.