To find the area bounded by the functions $f(x) = x + 3$ and $g(x) = e^{0.5x}$ between the lines $x = 0$ and $x = 2$, we need to set up and evaluate the definite integral of the difference between these two functions over the interval from $x = 0$ to $x = 2$.

Step 1: Set up the Integral

The area $A$ between two curves $f(x)$ and $g(x)$ over an interval $[a, b]$ is given by: $A = \int_{a}^{b} |f(x) - g(x)| \, dx$

In this case: $A = \int_{0}^{2} \left( (x + 3) - e^{0.5x} \right) \, dx$

Step 2: Evaluate the Integral

Let's break down this integral:

$A = \int_{0}^{2} (x + 3 - e^{0.5x}) \, dx$

We can evaluate this integral term by term.

1. Integral of $x$: $\int x \, dx = \frac{x^2}{2}$

2. Integral of 3: $\int 3 \, dx = 3x$

3. Integral of $e^{0.5x}$: $\int e^{0.5x} \, dx = \frac{e^{0.5x}}{0.5} = 2e^{0.5x}$

Putting it all together: $A = \left[ \frac{x^2}{2} + 3x - 2e^{0.5x} \right]_{0}^{2}$

Step 3: Substitute the Limits

Substitute $x = 2$ and $x = 0$ into the result:

1. For $x = 2$: $\frac{2^2}{2} + 3(2) - 2e^{0.5 \cdot 2} = 2 + 6 - 2e^{1} = 8 - 2e$

2. For $x = 0$: $\frac{0^2}{2} + 3(0) - 2e^{0.5 \cdot 0} = 0 + 0 - 2 = -2$

Now, subtract the result at $x = 0$ from the result at $x = 2$:

$A = (8 - 2e) - (-2) = 10 - 2e$

Step 4: Approximate to Two Decimal Places

Using the approximate value $e \approx 2.718$:

$A \approx 10 - 2 \times 2.718 = 10 - 5.436 = 4.56$

Final Answer

The area is approximately $\boxed{4.56}$.

Would you like more details on any of these steps?

Related Questions for Further Practice:

1. What is the area between $f(x) = x^2$ and $g(x) = \sin(x)$ over the interval $[0, \pi]$?
2. How would the area calculation change if $f(x) = e^{0.5x}$ and $g(x) = x + 3$ were reversed?
3. How do you find the area between two curves when they intersect multiple times within the given interval?
4. What is the geometric interpretation of integrating the absolute value of the difference between two functions?
5. How would you set up the integral for the area if the interval were different, such as from $x = 1$ to $x = 3$?

Tip:

Always check if the functions intersect within the interval, as this might require breaking the integral into multiple parts for an accurate area calculation.