# What is #int_(-4)^(1) |x^2-4|dx #?

Consider the function:

we know that a second polynomial with positive leading coefficient is negative in the interval between its roots and negative outside the interval, so:

Using the additivity of the definite integral we can split it as:

Now:

So:

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To evaluate the integral (\int_{-4}^{1} |x^2 - 4| , dx), we need to split the integral into intervals where (|x^2 - 4|) changes its behavior. Since (|x^2 - 4|) changes sign at (x = -2) and (x = 2), we partition the interval of integration accordingly:

(\int_{-4}^{1} |x^2 - 4| , dx = \int_{-4}^{-2} (4 - x^2) , dx + \int_{-2}^{2} (x^2 - 4) , dx + \int_{2}^{1} (x^2 - 4) , dx)

After integrating each part separately, we get:

(\int_{-4}^{1} |x^2 - 4| , dx = \frac{112}{3})

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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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