# What is the arclength of #f(x)=(1+x^2)/(x-1)# on #x in [2,3]#?

We get by the Quotient rule

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To find the arc length of the function ( f(x) = \frac{1+x^2}{x-1} ) on the interval ( [2,3] ), follow these steps:

- Compute the derivative of ( f(x) ), denoted ( f'(x) ).
- Use the formula for arc length: [ L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx ], where ( a = 2 ) and ( b = 3 ).
- Integrate ( \sqrt{1 + [f'(x)]^2} ) over the interval ( [2,3] ).
- The result of the integration is the arc length of ( f(x) ) on the interval ( [2,3] ).

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To find the arc length of ( f(x) = \frac{{1+x^2}}{{x-1}} ) on ( x ) in the interval ([2,3]), we can use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{{dy}}{{dx}}\right)^2} , dx ]

where (a = 2), (b = 3), and (y = f(x)). Then, we'll find the derivative of (f(x)) with respect to (x), which will give us ( \frac{{dy}}{{dx}} ), and then integrate.

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

[ \frac{{df}}{{dx}} = \frac{{d}}{{dx}}\left(\frac{{1+x^2}}{{x-1}}\right) ]

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

[ = \frac{{x^2 - 2x - 1 - x^2}}{{(x-1)^2}} ]

[ = \frac{{-2x - 1}}{{(x-1)^2}} ]

Now, we'll plug this into the formula for arc length and integrate:

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

This integral can be solved numerically using calculus techniques or numerical approximation methods.

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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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- How do you find the arc length of the curve #y=ln(cosx)# over the interval #[0,π/4]#?
- How much work does it take to pump the oil to the rim of the tank if the conical tank from #y=2x# is filled to within 2 feet of the top of the olive oil weighing #57(lb)/(ft)^3#?
- How do you find the arc length of the curve #y=e^(-x)+1/4e^x# from [0,1]?
- How do you find the volume of the solid generated by revolving the region bounded by the curves #y=x^(2)-x#, #y=3-x^(2)# rotated about the #y=4#?

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