What is the volume of the solid produced by revolving #f(x)=1/(x-1)-1/(x-2), x in [3,4] #around the x-axis?
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To find the volume of the solid produced by revolving the function ( f(x) = \frac{1}{x - 1} - \frac{1}{x - 2} ) on the interval ([3, 4]) around the x-axis, we use the disk method.
The volume of the solid of revolution can be calculated using the formula:
[ V = \pi \int_a^b [f(x)]^2 , dx ]
Where ( [a, b] ) is the interval of integration.
First, we square the function:
[ [f(x)]^2 = \left(\frac{1}{x - 1} - \frac{1}{x - 2}\right)^2 ]
Then we integrate this squared function from 3 to 4:
[ V = \pi \int_3^4 \left(\frac{1}{x - 1} - \frac{1}{x - 2}\right)^2 , dx ]
This integral can be calculated using appropriate techniques of integration or numerical methods.
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To find the volume of the solid produced by revolving the function (f(x) = \frac{1}{{x-1}} - \frac{1}{{x-2}}), where (x) is in the interval ([3,4]), around the x-axis, we can use the method of cylindrical shells.
The formula to find the volume of revolution using cylindrical shells is:
[V = 2\pi \int_a^b xf(x) , dx,]
where (a) and (b) are the limits of integration.
Substituting the given function (f(x) = \frac{1}{{x-1}} - \frac{1}{{x-2}}) into the formula, and the limits of integration (a = 3) and (b = 4), we have:
[V = 2\pi \int_3^4 x\left(\frac{1}{{x-1}} - \frac{1}{{x-2}}\right) , dx.]
Now, we simplify the integrand:
[V = 2\pi \int_3^4 \left( \frac{x}{{x-1}} - \frac{x}{{x-2}} \right) , dx.]
[V = 2\pi \left[ \int_3^4 \frac{x}{{x-1}} , dx - \int_3^4 \frac{x}{{x-2}} , dx \right].]
To evaluate each integral, we can use substitution. Let (u = x - 1) for the first integral, and (u = x - 2) for the second integral.
After integration and simplification, we'll have the volume of the solid produced by revolving (f(x)) around the x-axis over the interval ([3,4]).
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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.
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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