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The problem asks to find the area of the region enclosed by the curves \( y = e^{2x} \), \( y = e^{5x} \), between \( x = 0 \) and \( x = 1 \).

### Step-by-step Solution:

1. **Set up the integral:**

   To find the area between two curves, we use the formula:

   \[
   \text{Area} = \int_{a}^{b} \left[ f(x) - g(x) \right] \, dx
   \]

   Here, \( f(x) = e^{5x} \) is the upper function, and \( g(x) = e^{2x} \) is the lower function. The limits are from \( x = 0 \) to \( x = 1 \). Thus, the integral is:

   \[
   \text{Area} = \int_{0}^{1} \left( e^{5x} - e^{2x} \right) \, dx
   \]

2. **Compute the integral:**

   We now integrate each term separately:

   \[
   \int e^{5x} \, dx = \frac{e^{5x}}{5}, \quad \int e^{2x} \, dx = \frac{e^{2x}}{2}
   \]

   Therefore, the definite integral becomes:

   \[
   \text{Area} = \left[ \frac{e^{5x}}{5} - \frac{e^{2x}}{2} \right]_{0}^{1}
   \]

3. **Evaluate the integral:**

   Substituting the limits \( x = 1 \) and \( x = 0 \):

   At \( x = 1 \):

   \[
   \frac{e^{5(1)}}{5} - \frac{e^{2(1)}}{2} = \frac{e^5}{5} - \frac{e^2}{2}
   \]

   At \( x = 0 \):

   \[
   \frac{e^{5(0)}}{5} - \frac{e^{2(0)}}{2} = \frac{1}{5} - \frac{1}{2}
   \]

   So, the area is:

   \[
   \text{Area} = \left( \frac{e^5}{5} - \frac{e^2}{2} \right) - \left( \frac{1}{5} - \frac{1}{2} \right)
   \]

4. **Calculate the numeric value:**

   Now we will calculate the values for \( e^5 \), \( e^2 \), and perform the subtraction:

   \[
   \frac{e^5}{5} \approx \frac{148.413}{5} = 29.683
   \]
   \[
   \frac{e^2}{2} \approx \frac{7.389}{2} = 3.694
   \]
   \[
   \frac{1}{5} = 0.200, \quad \frac{1}{2} = 0.500
   \]

   Now substitute these values into the equation:

   \[
   \text{Area} \approx \left( 29.683 - 3.694 \right) - \left( 0.200 - 0.500 \right)
   \]
   \[
   \text{Area} \approx 25.989 + 0.300 = 26.289
   \]

   Thus, the area of the region is approximately **26.289 square units**.

---

Do you need more details on any of the steps? Here are 5 related questions you can explore further:

1. How do you determine which function is the upper and which is the lower in an area problem?
2. What is the general formula for finding the area between two curves?
3. How do you compute definite integrals for exponential functions like \( e^{2x} \) and \( e^{5x} \)?
4. How do you approximate values like \( e^2 \) and \( e^5 \) without a calculator?
5. Why do we subtract the lower bound from the upper bound when evaluating definite integrals?

**Tip:** Always ensure the limits of integration are carefully set based on where the curves intersect or are bounded within the given range.

Sketch the region enclosed by y = e^(2x), y = e^(5x), x = 0, and x = 1. Then find the area of the region.

Learn how to find the area of the region enclosed by the exponential functions y = e^(2x) and y = e^(5x) between x = 0 and x = 1. This problem involves calculus concepts like definite integrals and the area between curves, suitable for students in Grades 11-12.

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