# What is the area under the curve in the interval [-3, 3] for the function #f(x)=x^3-9x^2#?

I found

Consider the area (in orange):

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To find the area under the curve of the function ( f(x) = x^3 - 9x^2 ) in the interval ([-3, 3]), we can integrate the function over that interval using the definite integral formula:

[ \text{Area} = \int_{-3}^{3} (x^3 - 9x^2) , dx ]

Calculating the integral gives us:

[ \text{Area} = \left[ \frac{1}{4}x^4 - 3x^3 \right]_{-3}^{3} ]

Plugging in the upper and lower limits and subtracting gives:

[ \text{Area} = \left( \frac{1}{4}(3)^4 - 3(3)^3 \right) - \left( \frac{1}{4}(-3)^4 - 3(-3)^3 \right) ]

Simplifying further:

[ \text{Area} = \left( \frac{1}{4}(81) - 3(27) \right) - \left( \frac{1}{4}(81) + 3(27) \right) ]

[ \text{Area} = \left( \frac{81}{4} - 81 \right) - \left( \frac{81}{4} + 81 \right) ]

[ \text{Area} = -20.25 ]

Therefore, the area under the curve of the function ( f(x) = x^3 - 9x^2 ) in the interval ([-3, 3]) is (-20.25).

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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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