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The image shows a graph with a shaded region that appears to be bounded by two curves, \( y = x^2 \) and \( y = 4x - 10x^2 \), and the task is to find the area of this region.

To calculate the area between these two curves over a given interval, we follow these steps:

1. **Identify the points of intersection** of the curves \( y = x^2 \) and \( y = 4x - 10x^2 \) by setting them equal to each other:

   \[
   x^2 = 4x - 10x^2
   \]

2. **Simplify and solve** the resulting equation:

   \[
   11x^2 - 4x = 0
   \]
   \[
   x(11x - 4) = 0
   \]

   This gives the solutions \( x = 0 \) and \( x = \frac{4}{11} \).

3. **Set up the integral** to calculate the area between the curves from \( x = 0 \) to \( x = \frac{4}{11} \):

   \[
   \text{Area} = \int_{0}^{\frac{4}{11}} \left[(4x - 10x^2) - x^2\right] dx
   \]

4. **Simplify the integrand**:

   \[
   \text{Area} = \int_{0}^{\frac{4}{11}} \left(4x - 11x^2\right) dx
   \]

5. **Integrate** with respect to \( x \):

   \[
   \text{Area} = \left[ 2x^2 - \frac{11x^3}{3} \right]_{0}^{\frac{4}{11}}
   \]

6. **Evaluate** the definite integral:

   \[
   \text{Area} = \left(2\left(\frac{4}{11}\right)^2 - \frac{11\left(\frac{4}{11}\right)^3}{3}\right) - 0
   \]

After substituting the values, simplify the expression to find the exact area.

Would you like a detailed solution with all steps shown?

Here are five related questions:
1. What are other methods to find the intersection points of two curves?
2. How can you find the area between two curves if they intersect at multiple points?
3. What happens if the curves intersect over a different interval?
4. How do you interpret the result of an area calculation when it is negative?
5. Can you apply this method to curves defined by trigonometric functions?

**Tip:** When solving for the area between curves, always check the points of intersection carefully to ensure the correct limits of integration.

Learn how to calculate the area between curves y = x^2 and y = 4x - 10x^2 using integral calculus. Follow a step-by-step solution to find the exact area bounded by these curves from x = 0 to x = 4/11. Suitable for advanced high school students and calculus enthusiasts.

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