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

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To find the net area between ( f(x) = x^2 - \ln(x^2 + 1) ) and the x-axis over ( x ) in the interval ([1, 2]), we need to compute the definite integral of ( f(x) ) from 1 to 2.

[ \text{Net Area} = \int_{1}^{2} (x^2 - \ln(x^2 + 1)) , dx ]

After integrating, the net area will be the result.

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To find the net area between ( f(x) = x^2 - \ln(x^2 + 1) ) and the x-axis over ( x ) in the interval ([1, 2]), we need to evaluate the definite integral of ( |f(x)| ) over the given interval.

[ \text{Net area} = \int_{1}^{2} |f(x)| , dx ]

First, we need to determine the intervals where ( f(x) ) is above or below the x-axis over the interval ([1, 2]). We find this by examining the sign of ( f(x) ) for ( x ) in the interval.

Then, we calculate ( |f(x)| ) for each interval and integrate over those intervals separately. The net area will be the sum of the areas above the x-axis minus the areas below the x-axis.

Finally, we compute the definite integral of ( |f(x)| ) over each interval using appropriate integration techniques. The resulting net area will be the solution to the problem.

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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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