Key Concepts and Formulas

• Area Under a Curve: The area under a curve $y = f(x)$ from $x = a$ to $x = b$ is given by $\int_{a}^{b} f(x) \, dx$, provided $f(x) \geq 0$ on the interval $[a, b]$.
• Absolute Value: $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$.
• Integral Properties: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$ and $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$

Step-by-Step Solution

Step 1: Understanding the Region and the Function We are given the region defined by $-1 \leq x \leq 1$ and $0 \leq y \leq a + e^{|x|} - e^{-x}$, where $a > 0$. We need to find the value of $a$ such that the area of this region is $\frac{e^2 + 8e + 1}{e}$. The area is given by the integral of the function $f(x) = a + e^{|x|} - e^{-x}$ from $x = -1$ to $x = 1$. Since $a>0$ and $e^{|x|} > 0$ for all $x$, and $e^{-x}>0$, we need to verify that $f(x) \ge 0$ on $[-1,1]$.

Step 2: Dealing with the Absolute Value The presence of $|x|$ requires us to split the integral into two parts: one for $x < 0$ and one for $x \geq 0$. This is because the definition of $|x|$ changes at $x = 0$.

Step 3: Defining $f(x)$ for Each Interval

• For $x \in [-1, 0)$, $|x| = -x$, so $f(x) = a + e^{-x} - e^{-x} = a$.
• For $x \in [0, 1]$, $|x| = x$, so $f(x) = a + e^x - e^{-x}$.

Step 4: Setting up the Integral The total area $A$ is given by: $A = \int_{-1}^{1} (a + e^{|x|} - e^{-x}) \, dx = \int_{-1}^{0} (a + e^{|x|} - e^{-x}) \, dx + \int_{0}^{1} (a + e^{|x|} - e^{-x}) \, dx$ Substituting the expressions for $f(x)$ in each interval: $A = \int_{-1}^{0} a \, dx + \int_{0}^{1} (a + e^x - e^{-x}) \, dx$

Step 5: Evaluating the Integrals

• First Integral: $\int_{-1}^{0} a \, dx = [ax]_{-1}^{0} = a(0) - a(-1) = a$
• Second Integral: $\int_{0}^{1} (a + e^x - e^{-x}) \, dx = [ax + e^x + e^{-x}]_{0}^{1} = (a(1) + e^1 + e^{-1}) - (a(0) + e^0 + e^{-0}) = a + e + e^{-1} - (0 + 1 + 1) = a + e + e^{-1} - 2$

Step 6: Calculating the Total Area The total area is the sum of the two integrals: $A = a + (a + e + e^{-1} - 2) = 2a + e + e^{-1} - 2$

Step 7: Equating to the Given Area We are given that $A = \frac{e^2 + 8e + 1}{e} = e + 8 + e^{-1}$. Equating the two expressions for $A$: $2a + e + e^{-1} - 2 = e + 8 + e^{-1}$

Step 8: Solving for $a$ Subtracting $e + e^{-1}$ from both sides, we get: $2a - 2 = 8$ Adding 2 to both sides gives: $2a = 10$ Dividing by 2 gives: $a = 5$

Common Mistakes & Tips

• Remember to split the integral when dealing with absolute values. Forgetting to do so will lead to an incorrect result.
• Simplify the integrand as much as possible before integrating. In this case, $a + e^{|x|} - e^{-x}$ simplifies to $a$ for $x \in [-1, 0)$.
• Be careful with the sign when integrating $e^{-x}$. The integral is $-e^{-x}$.

Summary We found the value of $a$ by splitting the integral due to the absolute value, simplifying the integrand, evaluating the definite integrals, and equating the result to the given area. Solving for $a$ yielded $a=5$.

The final answer is \boxed{5}, which corresponds to option (B).