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Hallar el aérea de la región sombreada de las figuras geométricas

To find the area of the shaded region, we'll consider the area of each section individually. The region consists of several rectangular and potentially non-rectangular shapes, determined by the provided function expressions.

1. **Top-left Rectangle:**
   - Height: \( 2x \)
   - Width: \( x^3 - 4 \)
   - Area: \( A_1 = 2x \times (x^3 - 4) \)

2. **Top-middle Rectangle:**
   - Height: \( 2x \)
   - Width: \( x^3 - 2 - (x^3 - 4) = 2 \)
   - Area: \( A_2 = 2x \times 2 \)

3. **Middle Rectangle:**
   - Height: \( x^3 - 2 \)
   - Width: \( 2x \)
   - Area: \( A_3 = (x^3 - 2) \times 2x \)

4. **Bottom-right Rectangle:**
   - Height: \( x^3 - 2 \)
   - Width: \( 2x \)
   - Area: \( A_4 = (x^3 - 2) \times 2x \)

5. **Bottom-right Square:**
   - Side: \( x^3 \)
   - Area: \( A_5 = (x^3)^2 \)

The total area \( A \) of the shaded region is the sum of these areas:

\[
A = A_1 + A_2 + A_3 + A_4 + A_5
\]

Let's calculate each area and then sum them up.The total area of the shaded region is given by the expression:

\[
A = x(x^5 + 6x^3 - 12)
\]

Would you like to explore any specific aspects of this calculation, such as verifying individual area contributions, or have any other questions?

Here are some other questions you might consider:
1. How can this expression be simplified further?
2. What are the roots of the polynomial in the area expression?
3. How does the area change with different values of \( x \)?
4. What if the expressions defining the rectangles changed?
5. How would you calculate the area if the shapes were curved instead of straight?

**Tip:** When dealing with composite shapes, it's often helpful to break them down into simpler geometric forms to calculate areas or volumes more easily.

Discover how to calculate the area of shaded geometric figures using mathematical formulas and expressions. This problem involves composite shapes made up of rectangles and squares, with varying dimensions based on algebraic expressions. Suitable for advanced high school students in Grades 10-12.

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