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The problem is asking to determine the area under the curve for the function:

\[
y = x^3 - 2x^2 - x + 2
\]

between \(x = -1\) and \(x = 2\). This is done by calculating the definite integral of the function from \(x = -1\) to \(x = 2\).

To solve for the area, we need to evaluate the following integral:

\[
\text{Area} = \int_{-1}^{2} (x^3 - 2x^2 - x + 2) \, dx
\]

Let's compute the integral and determine the area.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the steps to manually calculate the integral if you'd like. Would you like to proceed with that explanation?

Here are some related questions that might be helpful:
1. How do you manually integrate a polynomial function?
2. What does the definite integral represent in terms of area?
3. How do you handle negative areas in definite integrals?
4. Why is the area under the curve important in real-world applications?
5. What methods can you use to verify your integral calculations?

**Tip:** When solving definite integrals, always consider the symmetry of the curve and whether parts of it are below the x-axis, as this affects the total area.

Determine the area between the graph of the function y = x^3 - 2x^2 - x + 2 and the x-axis for x ∈ [-1, 2].

Learn how to calculate the area under the curve for the function y = x^3 - 2x^2 - x + 2 between x = -1 and x = 2 using the definite integral. This problem covers integral calculus concepts and applies the Fundamental Theorem of Calculus. Suitable for students in Grades 11-12.

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