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Find the area bounded by the curves and lines

The problem asks for the area bounded by the curve \( y = e^{2x} \), the line \( y = 3 \), and the y-axis \( x = 0 \).

We can approach this by calculating the area between the curve and the line, from \( x = 0 \) to the value of \( x \) where \( y = e^{2x} = 3 \).

### Steps to find the area:

1. **Find the intersection of \( y = e^{2x} \) and \( y = 3 \):**
   Set \( e^{2x} = 3 \) and solve for \( x \).

   \[
   e^{2x} = 3 \implies 2x = \ln(3) \implies x = \frac{\ln(3)}{2}
   \]

   So, the curves intersect at \( x = \frac{\ln(3)}{2} \).

2. **Set up the integral:**
   The area between the curves is the integral of the difference between the upper function \( y = 3 \) and the lower function \( y = e^{2x} \), from \( x = 0 \) to \( x = \frac{\ln(3)}{2} \).

   The area is given by:
   \[
   A = \int_0^{\frac{\ln(3)}{2}} \left( 3 - e^{2x} \right) \, dx
   \]

3. **Compute the integral:**

   Let's calculate the area step by step:
   \[
   \int_0^{\frac{\ln(3)}{2}} 3 \, dx = 3 \cdot \frac{\ln(3)}{2} = \frac{3 \ln(3)}{2}
   \]

   For the second term:
   \[
   \int_0^{\frac{\ln(3)}{2}} e^{2x} \, dx = \frac{1}{2} \left[ e^{2x} \right]_0^{\frac{\ln(3)}{2}} = \frac{1}{2} \left( e^{\ln(3)} - e^0 \right) = \frac{1}{2} \left( 3 - 1 \right) = \frac{1}{2} \times 2 = 1
   \]

4. **Final area calculation:**
   \[
   A = \frac{3 \ln(3)}{2} - 1
   \]

Thus, the area bounded by the curves is \( \frac{3 \ln(3)}{2} - 1 \).

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Would you like further explanation or any additional details?

Here are five related questions to deepen your understanding:
1. How does the natural logarithm \( \ln(x) \) help in solving equations involving exponentials?
2. How do you determine whether to subtract one curve from another when calculating area?
3. What changes if the upper curve or line is different, such as \( y = 4 \) instead of \( y = 3 \)?
4. How would you find the area if the curve were \( y = e^{x} \) instead of \( y = e^{2x} \)?
5. How does changing the bounds of integration affect the calculated area?

**Tip:** When solving for the area between curves, always ensure you correctly identify which curve is above the other over the integration interval.

Find the area bounded by the curves and lines y = e^(2x), y = 3, and x = 0 (y-axis).

This problem involves finding the area bounded by the curve y = e^(2x), the line y = 3, and the y-axis (x = 0). The solution includes calculating the definite integral between these curves and involves exponential functions and logarithms. Suitable for students in Grades 11-12.

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