It seems there was an issue solving the equation directly for $b$. Let's solve the problem step-by-step.

We need to solve the integral: $\int_{-b}^{b} |x - 4| \, dx = 47$

Step-by-Step Solution

1. Understanding the Absolute Value Function: The function $|x - 4|$ has a critical point at $x = 4$. This divides the integral into two parts:

   \begin{cases} 4 - x & \text{if } x < 4 \\ x - 4 & \text{if } x \geq 4 \end{cases}$$

2. Case Analysis for the Integral:

   • If $b \leq 4$: $\int_{-b}^{b} |x - 4| \, dx = \int_{-b}^{b} (4 - x) \, dx$ This case doesn't match the given integral result since the integral value is quite large (47), suggesting $b$ is greater than 4.

   • If $b > 4$: $\int_{-b}^{b} |x - 4| \, dx = \int_{-b}^{4} (4 - x) \, dx + \int_{4}^{b} (x - 4) \, dx$

3. Evaluating the Integral:

   • For $\int_{-b}^{4} (4 - x) \, dx$: $\int_{-b}^{4} (4 - x) \, dx = \left[ 4x - \frac{x^2}{2} \right]_{-b}^{4}$ $= \left( 4 \cdot 4 - \frac{4^2}{2} \right) - \left( 4(-b) - \frac{(-b)^2}{2} \right)$ $= (16 - 8) - (-4b - \frac{b^2}{2})$ $= 8 + 4b + \frac{b^2}{2}$

   • For $\int_{4}^{b} (x - 4) \, dx$: $\int_{4}^{b} (x - 4) \, dx = \left[ \frac{x^2}{2} - 4x \right]_{4}^{b}$ $= \left( \frac{b^2}{2} - 4b \right) - \left( \frac{4^2}{2} - 4 \cdot 4 \right)$ $= \frac{b^2}{2} - 4b - (8 - 16)$ $= \frac{b^2}{2} - 4b + 8$

4. Combining Both Parts: $\int_{-b}^{b} |x - 4| \, dx = \left( 8 + 4b + \frac{b^2}{2} \right) + \left( \frac{b^2}{2} - 4b + 8 \right)$ $= 8 + 8 + 4b - 4b + \frac{b^2}{2} + \frac{b^2}{2}$ $= 16 + b^2$

5. Setting Up the Equation: $16 + b^2 = 47$

6. Solving for $b^2$: $b^2 = 47 - 16$ $b^2 = 31$

Therefore, the value of $b^2$ is $31$.

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you integrate absolute value functions in general?
2. What are the properties of definite integrals involving absolute value functions?
3. Can you explain the geometric interpretation of the integral of $|x - 4|$?
4. How would the solution change if the limits of integration were different?
5. What if the integrand was $|x - c|$ for some constant $c$? How would you approach the problem?

Tip: When dealing with absolute value integrals, always identify the points where the expression inside the absolute value changes sign. This helps in splitting the integral into manageable parts.