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The image shows a problem involving the graph of the function \( y = x(x^2 - 2) \), with the goal of finding the value of \( a \) such that the areas of the two shaded regions (one below and one above the x-axis) are equal. The graph shows two areas, one below the x-axis between 0 and \( a \), and another one above the x-axis between \( a \) and some other point. The equation \( a = 2 \) is already provided as the solution.

Let me explain the solution in detail:

### Steps to Solve:

1. **Function Analysis**:
   The function is given as \( y = x(x^2 - 2) = x^3 - 2x \). This is a cubic function with roots at \( x = 0 \) and \( x = \pm \sqrt{2} \), where the function crosses the x-axis.

2. **Determine the areas**:
   The goal is to find the value of \( a \) such that the areas of the shaded regions below and above the x-axis are equal. These areas are determined by integrating the absolute value of the function over the respective intervals.

3. **Set up the integrals**:
   - For the region below the x-axis (from 0 to \( a \)): The function is negative in this interval, so we calculate the area as:
     \[
     A_1 = \int_0^a - (x^3 - 2x) \, dx
     \]
   
   - For the region above the x-axis (from \( a \) to \( b \)): The function is positive here, so the area is:
     \[
     A_2 = \int_a^b (x^3 - 2x) \, dx
     \]

4. **Find the points where the function crosses the x-axis**:
   The function crosses the x-axis at \( x = -\sqrt{2}, 0, \sqrt{2} \).

5. **Use symmetry**:
   The function \( x^3 - 2x \) is symmetric about the origin, meaning the area above the x-axis and below the x-axis will be equal when \( a = 2 \).

Thus, the solution is \( a = 2 \), which divides the areas of the shaded regions equally.

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Would you like more details or have any questions?

Here are 5 questions related to this explanation:

1. How do we calculate the definite integral of \( x^3 - 2x \) over a specific interval?
2. What is the geometric significance of finding the roots of a cubic function?
3. How does symmetry play a role in determining equal areas for this function?
4. Can you explain how to set up the limits of integration for different intervals?
5. What changes would occur if we shifted or stretched the given function?

**Tip:** Always check the sign of the function when calculating areas between the curve and the x-axis to determine whether the area is positive or negative.

Find the value of a such that the 2 shaded regions have equal areas for the function y = x(x^2 - 2). The solution provided states a = 2.

Explore how to find the value of 'a' where the areas of two shaded regions formed by the cubic function y = x(x^2 - 2) are equal. This calculus problem involves integration and symmetry. Solution shows that a = 2.

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