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We are tasked with finding the area of the region enclosed by the curves \( y = e^{5x} \), \( y = e^{10x} \), and the vertical line \( x = 1 \).

### Step-by-step Solution:

1. **Identify the boundaries:**
   - The two curves are \( y = e^{5x} \) and \( y = e^{10x} \).
   - The region is bound between these two curves, and from \( x = 0 \) (since both exponentials are zero at \( x = 0 \)) to \( x = 1 \) as given by the vertical line.

2. **Set up the integral for the area:**
   The area between two curves from \( x = a \) to \( x = b \) is given by the integral:
   \[
   A = \int_a^b [f(x) - g(x)] \, dx
   \]
   where \( f(x) \) is the upper curve and \( g(x) \) is the lower curve.

   In this case:
   - The upper curve is \( y = e^{10x} \)
   - The lower curve is \( y = e^{5x} \)
   - The bounds are \( x = 0 \) to \( x = 1 \)

   So the area \( A \) is:
   \[
   A = \int_0^1 (e^{10x} - e^{5x}) \, dx
   \]

3. **Solve the integral:**

   \[
   A = \int_0^1 e^{10x} \, dx - \int_0^1 e^{5x} \, dx
   \]

   We integrate each exponential term separately:
   - \( \int e^{10x} \, dx = \frac{1}{10} e^{10x} \)
   - \( \int e^{5x} \, dx = \frac{1}{5} e^{5x} \)

   So, evaluating from \( 0 \) to \( 1 \):

   \[
   A = \left[ \frac{1}{10} e^{10x} \right]_0^1 - \left[ \frac{1}{5} e^{5x} \right]_0^1
   \]

   Substituting the bounds:

   \[
   A = \left( \frac{1}{10} e^{10} - \frac{1}{10} e^0 \right) - \left( \frac{1}{5} e^5 - \frac{1}{5} e^0 \right)
   \]

   Simplifying further:

   \[
   A = \frac{1}{10} (e^{10} - 1) - \frac{1}{5} (e^5 - 1)
   \]

4. **Combine the terms:**

   \[
   A = \frac{1}{10} e^{10} - \frac{1}{10} - \frac{1}{5} e^5 + \frac{1}{5}
   \]

   \[
   A = \frac{1}{10} e^{10} - \frac{1}{5} e^5 + \left( \frac{1}{5} - \frac{1}{10} \right)
   \]

   Simplifying the constants:

   \[
   A = \frac{1}{10} e^{10} - \frac{1}{5} e^5 + \frac{1}{10}
   \]

Thus, the area enclosed by the curves is:

\[
A = \frac{1}{10} e^{10} - \frac{1}{5} e^5 + \frac{1}{10}
\]

### Conclusion:
This expression gives the exact area. You can also compute the approximate value by substituting the values of \( e^{10} \) and \( e^{5} \).

Would you like to compute the approximate value of this area or have any clarifications? 

---

### Relative Questions:
1. How does changing the exponent in an exponential function affect its graph?
2. What is the general formula for finding the area between two curves?
3. How do you calculate definite integrals for exponential functions?
4. Why is the exponential function \( e^x \) important in calculus?
5. What happens to the enclosed area if we change the boundary at \( x = 1 \) to another value?

### Tip:
When working with exponential functions, the difference between large and small exponents can lead to dramatic changes in the function’s behavior, especially over larger intervals.

Sketch the region enclosed by y = e^(5x), y = e^(10x), and x = 1. Find the area of the region.

This problem explores finding the area between the exponential curves y = e^(5x) and y = e^(10x) from x = 0 to x = 1 using definite integration. Learn how to set up the integral and solve it step-by-step.

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