Home > Calculus > Determining the Surface Area of a Solid of Revolution

What is the surface area produced by rotating #f(x)=1/x-1/(x+3), x in [1,3]# around the x-axis?

Answer 1

Scarlett Allison

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Answer 2

Scarlett Allison

To find the surface area produced by rotating the function ( f(x) = \frac{1}{x} - \frac{1}{x + 3} ) over the interval ( x ) in [1,3] around the x-axis, you can use the formula for surface area of revolution:

[ S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left( f'(x) \right)^2} , dx ]

Where ( f'(x) ) is the derivative of ( f(x) ).

First, find ( f'(x) ) by taking the derivative of ( f(x) ):

[ f'(x) = -\frac{1}{x^2} + \frac{1}{(x + 3)^2} ]

Now, plug ( f(x) ) and ( f'(x) ) into the surface area formula:

[ S = 2\pi \int_{1}^{3} \left( \frac{1}{x} - \frac{1}{x + 3} \right) \sqrt{1 + \left( -\frac{1}{x^2} + \frac{1}{(x + 3)^2} \right)^2} , dx ]

Finally, evaluate the integral to find the surface area.

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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.