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The problem presents a geometric figure where the area of the circular trapezoid DFBE is six times the area of the circle inscribed at point D, and we are tasked with calculating the expression \( E = 3\pi \sin \theta + 2\theta \).

### Let’s break down the steps to solve this problem:

1. **Area of the Circle:**
   The circle is inscribed at point D with radius \( r \). The area \( A_{\text{circle}} \) of a circle is given by:
   \[
   A_{\text{circle}} = \pi r^2
   \]

2. **Area of the Circular Trapezoid DFBE:**
   The circular trapezoid DFBE can be thought of as the difference between the area of the quarter circle with radius \( OB \) (which is \( R \)) and the area of triangle \( OEB \).

   - The area of the quarter circle is:
     \[
     A_{\text{quarter circle}} = \frac{1}{4} \pi R^2
     \]
   
   - The area of triangle \( OEB \) can be found using trigonometry or geometry, depending on additional known values.

   According to the problem, the area of DFBE is six times the area of the circle, so:
   \[
   A_{\text{DFBE}} = 6 A_{\text{circle}} = 6 \pi r^2
   \]

3. **Setting up the equation:**
   From the given condition, the area of DFBE can be expressed as:
   \[
   A_{\text{DFBE}} = \frac{1}{4} \pi R^2 - A_{\text{triangle OEB}}
   \]
   Setting this equal to \( 6 \pi r^2 \) gives us the equation:
   \[
   \frac{1}{4} \pi R^2 - A_{\text{triangle OEB}} = 6 \pi r^2
   \]
   This will allow us to solve for \( R \) in terms of \( r \), and then apply further trigonometric identities to solve for \( E = 3\pi \sin \theta + 2\theta \).

I will continue solving this if you can confirm the exact values of \( \theta \) or any specific data for triangle \( OEB \). Would you like to proceed or need more clarifications?

Here are some questions that might help you understand or extend this concept:
1. How is the area of a circular trapezoid generally calculated?
2. What is the relationship between the radius of the circle and the larger quarter circle in this context?
3. How does the angle \( \theta \) affect the areas in the figure?
4. Why is the factor of 6 important in the comparison of areas?
5. How can trigonometry help in calculating unknown lengths in the triangle \( OEB \)?

**Tip:** Whenever working with areas of segments or portions of circles, it's useful to remember formulas for both full circles and fractions of a circle (like half or quarter circles).

En la figura mostrada, se cumple que el área del trapecio circular DFBE es 6 veces el área del círculo. Calcule E = 3πsenθ + 2θ.

In this problem, we explore the relationship between the area of a circular trapezoid and an inscribed circle. The task is to calculate the expression E = 3πsinθ + 2θ, where the area of the trapezoid DFBE is 6 times the area of the inscribed circle. This involves geometric formulas for circle areas and trigonometric identities.

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