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To solve this problem, we need to compute the area enclosed by the curves \( f(x) = e^x \) and \( g(x) = x \) over the interval \([0, 3]\).

### Steps to solve:
1. **Identify the curves**: 
   - \( f(x) = e^x \)
   - \( g(x) = x \)

2. **Find the points of intersection**:
   We need to check if the functions intersect in the given interval. Set \( e^x = x \) and solve for \(x\). From numerical or graphical methods, it's known that \( e^x \) and \( x \) do not intersect between \( [0, 3] \). Therefore, we can skip finding the exact point of intersection because \( e^x > x \) throughout this interval.

3. **Set up the integral**:
   Since \( f(x) = e^x \) is above \( g(x) = x \) over the interval \([0, 3]\), the area between the curves is given by:
   \[
   A = \int_0^3 \left( e^x - x \right) \, dx
   \]

4. **Compute the integral**:
   We will evaluate:
   \[
   A = \int_0^3 e^x \, dx - \int_0^3 x \, dx
   \]

   The integral of \( e^x \) is \( e^x \), and the integral of \( x \) is \( \frac{x^2}{2} \). Now let's calculate this.

   \[
   A = \left[ e^x \right]_0^3 - \left[ \frac{x^2}{2} \right]_0^3
   \]

5. **Substitute and evaluate**:
   - \( \left[ e^x \right]_0^3 = e^3 - e^0 = e^3 - 1 \)
   - \( \left[ \frac{x^2}{2} \right]_0^3 = \frac{3^2}{2} - \frac{0^2}{2} = \frac{9}{2} = 4.5 \)

   Therefore, the area is:
   \[
   A = (e^3 - 1) - 4.5
   \]

Let me calculate this to four decimal places.The area enclosed by the curves \( f(x) = e^x \) and \( g(x) = x \) on the interval \([0, 3]\) is approximately **14.5855** (rounded to four decimal places).

Would you like any further details or explanations?

Here are some related questions you might find useful:

1. How do you determine which function is on top in a given region?
2. What are some methods to find the intersection points of curves?
3. How would the problem change if the interval were different, such as \([1, 4]\)?
4. How do you compute the area between curves when they intersect at multiple points?
5. How can you use numerical methods to approximate areas under curves?

**Tip**: Always check whether functions intersect in the given domain to ensure you're calculating the correct area between them.

Find the area of the region enclosed by f(x) = e^x and g(x) = x on the domain [0, 3].

Learn how to calculate the area between the curves f(x) = e^x and g(x) = x over the interval [0, 3] using integration. This problem involves advanced calculus concepts such as the Fundamental Theorem of Calculus. Suitable for high school and college-level students.

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