# How do you evaluate the definite integral #int (x-2) dx# from [-1,0]?

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To evaluate the definite integral (\int_{-1}^{0} (x-2) , dx), you need to find the antiderivative of the function (x-2) and then evaluate it at the upper and lower limits of integration, and then subtract the result at the lower limit from the result at the upper limit.

The antiderivative of (x-2) is (\frac{1}{2}x^2 - 2x + C), where (C) is the constant of integration.

Evaluate the antiderivative at the upper limit ((0)) and the lower limit ((-1)), then subtract:

(\left[\frac{1}{2}(0)^2 - 2(0)\right] - \left[\frac{1}{2}(-1)^2 - 2(-1)\right])

Simplify:

(\left[0 - 0\right] - \left[\frac{1}{2} - (-2)\right])

(-\frac{1}{2} + 2)

(= \frac{3}{2})

Therefore, (\int_{-1}^{0} (x-2) , dx = \frac{3}{2}).

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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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