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To find the arc length of the curve \( y = 2x^{3/2} \) on the interval \( 1 \leq x \leq 3 \), we use the formula for arc length:

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

### Step 1: Find \( \frac{dy}{dx} \)
The given function is \( y = 2x^{3/2} \). To find the derivative, apply the power rule:

\[
\frac{dy}{dx} = 2 \cdot \frac{3}{2} x^{1/2} = 3x^{1/2}
\]

### Step 2: Square the derivative
\[
\left( \frac{dy}{dx} \right)^2 = (3x^{1/2})^2 = 9x
\]

### Step 3: Set up the arc length integral
Now substitute \( \left( \frac{dy}{dx} \right)^2 = 9x \) into the arc length formula:

\[
L = \int_{1}^{3} \sqrt{1 + 9x} \, dx
\]

### Step 4: Solve the integral
The integral we need to solve is:

\[
L = \int_{1}^{3} \sqrt{9x + 1} \, dx
\]

We can use a simple substitution method to solve this. Let:

\[
u = 9x + 1 \quad \Rightarrow \quad du = 9 \, dx
\]

So, when \( x = 1 \), \( u = 10 \), and when \( x = 3 \), \( u = 28 \). The integral becomes:

\[
L = \frac{1}{9} \int_{10}^{28} \sqrt{u} \, du
\]

This is now straightforward to integrate:

\[
L = \frac{1}{9} \cdot \frac{2}{3} u^{3/2} \Big|_{10}^{28}
\]

\[
L = \frac{2}{27} \left( 28^{3/2} - 10^{3/2} \right)
\]

### Step 5: Calculate the final result
Now, let's compute the values of \( 28^{3/2} \) and \( 10^{3/2} \):

\[
28^{3/2} = 28 \times \sqrt{28} \approx 28 \times 5.2915 = 148.162
\]
\[
10^{3/2} = 10 \times \sqrt{10} \approx 10 \times 3.162 = 31.622
\]

Thus,

\[
L = \frac{2}{27} \left( 148.162 - 31.622 \right) = \frac{2}{27} \times 116.54 \approx 8.63
\]

The arc length of the curve \( y = 2x^{3/2} \) from \( x = 1 \) to \( x = 3 \) is approximately \( 8.63 \) units.

Would you like more details or have any questions?

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**Related Questions:**
1. How do you derive the formula for arc length in general?
2. What is the significance of the derivative in calculating arc length?
3. How would the arc length change if the limits were different, say from \( x = 2 \) to \( x = 4 \)?
4. Can this method be applied to parametric equations for finding arc length?
5. How can substitution methods simplify complex integrals like this one?

**Tip:** Always check for symmetry in problems involving arc length to potentially reduce computation complexity.

Find the arc length of y = 2x^(3/2) on 1 <= x <= 3.

Learn how to find the arc length of the curve y = 2x^(3/2) between x = 1 and x = 3. This solution covers key calculus concepts, including the arc length formula, differentiation, and substitution methods for integration.

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