Let's break down the problem step-by-step:

Given: $g(x) = \int_{-2}^{2} |x - t| (t^2 - a^2) \, dt,$ we need to find the range of $a$ such that $g(x)$ has its minimum value, and also express this minimum value in terms of $a$.

Step-by-step Analysis:

1. Understanding $g(x)$: $g(x)$ involves an integral from $-2$ to $2$ of the product of two functions:

   • $|x - t|$, which represents the absolute difference between $x$ and $t$.
   • $(t^2 - a^2)$, a quadratic function that depends on the parameter $a$.

2. Behavior of the Integral:

   • The function $|x - t|$ changes its behavior depending on whether $t < x$ or $t \geq x$.
   • The function $(t^2 - a^2)$ changes sign depending on whether $|t| < |a|$ or $|t| \geq |a|$.

3. Split the Integral: Since $|x - t|$ can be split into different cases:

   • When $t < x$: $|x - t| = x - t$.
   • When $t \geq x$: $|x - t| = t - x$.

   Depending on $x$, split the integral: $g(x) = \int_{-2}^{x} (x - t)(t^2 - a^2) \, dt + \int_{x}^{2} (t - x)(t^2 - a^2) \, dt.$

4. Finding the Minimum Value: To find the minimum value of $g(x)$, consider the integrand's symmetry and behavior as a function of $a$. The minimum value will depend on the sign of $(t^2 - a^2)$ and how it interacts with $|x - t|$.

5. Conditions on $a$:

   • If $a^2 < 4$, then $t^2 - a^2$ will be negative for some parts of the interval $[-2, 2]$.
   • If $a^2 \geq 4$, then $t^2 - a^2$ is non-positive for the entire range of integration.

Conclusion:

1. Finding the Range for $a$:
   • The range of $a$ for which $g(x)$ achieves its minimum is when $a^2 \geq 4$, or equivalently $|a| \geq 2$.
2. Expressing Minimum Value:
   • When $|a| \geq 2$, $g(x)$ reaches a minimum at some specific values of $x$ (likely by symmetry, possibly at $x = 0$).

Final Answer:

1. The range of $a$ such that $g(x)$ has its minimum value: $|a| \geq 2$.
2. Express $g(x)$'s minimum value in terms of $a$.

Let me know if you want to go deeper into the integration process or have any other questions!

Related Questions:

1. How do you handle integrals with absolute value functions?
2. What is the geometric interpretation of $g(x)$ in this context?
3. How does the choice of $a$ affect the sign of the integrand?
4. What is the role of symmetry in finding the minimum value of this integral?
5. How can you simplify the integral if $x = 0$?

Tip:

When dealing with integrals that include absolute values, always consider breaking the integral into different intervals where the absolute value expression can be simplified.