# What is the net area between #f(x) = -xln(x^2-1) # and the x-axis over #x in [2, 4 ]#?

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To find the net area between ( f(x) = -x\ln(x^2-1) ) and the x-axis over ( x ) in the interval ([2, 4]), you need to integrate the absolute value of ( f(x) ) over that interval and subtract the integral of ( f(x) ) itself over the same interval. The integral of ( f(x) ) represents the area below the x-axis, and subtracting it ensures that the net area is positive.

The net area ( A ) is given by: [ A = \int_{2}^{4} |f(x)| , dx - \int_{2}^{4} f(x) , dx ]

Substitute the function ( f(x) = -x\ln(x^2-1) ) into the equation and evaluate the integrals.

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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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