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To find the length of the graph of the function \( y = \frac{1}{3}(x^2 + 4)^{3/2} \) from \( x = -2 \) to \( x = 2 \), we'll use the arc length formula:

\[
L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]

### Step 1: Differentiate the function \( y(x) \)
Given:

\[
y(x) = \frac{1}{3}(x^2 + 4)^{3/2}
\]

We need to find \( \frac{dy}{dx} \).

First, apply the chain rule to differentiate \( y(x) \):

\[
\frac{dy}{dx} = \frac{1}{3} \cdot \frac{3}{2} (x^2 + 4)^{\frac{3}{2} - 1} \cdot 2x = x(x^2 + 4)^{\frac{1}{2}}
\]

### Step 2: Plug \( \frac{dy}{dx} \) into the arc length formula
The arc length formula becomes:

\[
L = \int_{-2}^{2} \sqrt{1 + \left( x(x^2 + 4)^{\frac{1}{2}} \right)^2} \, dx
\]

Simplify the expression inside the square root:

\[
L = \int_{-2}^{2} \sqrt{1 + x^2(x^2 + 4)} \, dx
\]

\[
L = \int_{-2}^{2} \sqrt{1 + x^4 + 4x^2} \, dx
\]

\[
L = \int_{-2}^{2} \sqrt{x^4 + 4x^2 + 1} \, dx
\]

### Step 3: Evaluate the integral
To express the integral in exact form, recognize that:

\[
x^4 + 4x^2 + 1 = (x^2 + 2)^2
\]

So,

\[
L = \int_{-2}^{2} \sqrt{(x^2 + 2)^2} \, dx
\]

This simplifies to:

\[
L = \int_{-2}^{2} |x^2 + 2| \, dx
\]

Since \( x^2 + 2 \) is always positive, the absolute value can be removed:

\[
L = \int_{-2}^{2} (x^2 + 2) \, dx
\]

Evaluate the integral:

\[
L = \int_{-2}^{2} x^2 \, dx + \int_{-2}^{2} 2 \, dx
\]

The first integral:

\[
\int x^2 \, dx = \frac{x^3}{3} \Big|_{-2}^{2} = \left(\frac{8}{3} - \frac{-8}{3}\right) = \frac{16}{3}
\]

The second integral:

\[
\int 2 \, dx = 2x \Big|_{-2}^{2} = 2(2 - (-2)) = 8
\]

So, the total length \( L \) is:

\[
L = \frac{16}{3} + 8 = \frac{16}{3} + \frac{24}{3} = \frac{40}{3}
\]

Thus, the exact length of the curve is:

\[
L = \frac{40}{3}
\]

### Final Answer:
The length of the graph of the function from \( x = -2 \) to \( x = 2 \) is \( \frac{40}{3} \) units.

---

Would you like any further details on this solution, or have any questions? Here are some related questions you might explore:

1. How does the arc length formula change for different types of functions?
2. What would the length of the curve be if the limits of integration were from \( x = 0 \) to \( x = 2 \)?
3. Can the arc length formula be applied to parametric equations, and if so, how?
4. How does the arc length formula adapt to polar coordinates?
5. What are the potential challenges when evaluating arc lengths for more complex functions?

**Tip:** When solving problems involving arc lengths, always check if the function simplifies after differentiation; this can make evaluating the integral much easier.

Use the arc-length formula to find the length of the graph of the function y = (1/3)(x^2 + 4)^(3/2) from x = -2 to x = 2. (Express numbers in exact form. Use symbolic notation and fractions where needed.)

Learn how to find the length of the graph of the function y = (1/3)(x^2 + 4)^(3/2) from x = -2 to x = 2 using the arc length formula. This problem covers key calculus concepts such as differentiation and integration, suitable for high school and college students.

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