Set-up the integral to calculate the length of the curve  y = [(x^3) / 2] + [1 / (6x)] for 3 ≤ x ≤ 5.  Evaluate the integral.

To find the length of the curve \( y = \frac{x^3}{2} + \frac{1}{6x} \) over the interval \( 3 \leq x \leq 5 \), we proceed with the following steps.

### **Step 1: Recall the formula for the length of a curve**

The length \( L \) of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is given by:
\[
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]

### **Step 2: Find the derivative \(\frac{dy}{dx}\)**

Given:
\[
y = \frac{x^3}{2} + \frac{1}{6x}
\]
Differentiate \( y \) with respect to \( x \):
\[
\frac{dy}{dx} = \frac{3x^2}{2} - \frac{1}{6x^2}
\]

### **Step 3: Set up the integral**

The formula for the arc length becomes:
\[
L = \int_{3}^{5} \sqrt{1 + \left(\frac{3x^2}{2} - \frac{1}{6x^2}\right)^2} \, dx
\]

### **Step 4: Simplify the expression inside the square root**

First, let’s expand \( \left(\frac{3x^2}{2} - \frac{1}{6x^2}\right)^2 \):
\[
\left(\frac{3x^2}{2} - \frac{1}{6x^2}\right)^2 = \left(\frac{9x^4}{4} - \frac{x^2}{2} + \frac{1}{36x^4}\right)
\]
Thus,
\[
1 + \left(\frac{3x^2}{2} - \frac{1}{6x^2}\right)^2 = 1 + \frac{9x^4}{4} - \frac{x^2}{2} + \frac{1}{36x^4}
\]

The integral to evaluate is:
\[
L = \int_{3}^{5} \sqrt{1 + \frac{9x^4}{4} - \frac{x^2}{2} + \frac{1}{36x^4}} \, dx
\]

Now, let's evaluate this integral numerically.

### **Step 5: Numerical Evaluation**

Let's proceed with calculating this integral.The length of the curve \( y = \frac{x^3}{2} + \frac{1}{6x} \) over the interval \( 3 \leq x \leq 5 \) is approximately \( 48.71 \) units.

Would you like further details or have any questions?

### Related Questions
1. How would the arc length change if the interval were different?
2. What is the geometric interpretation of the derivative in the arc length formula?
3. How would you set up the integral if the curve were given parametrically?
4. How do arc length calculations change when working in polar coordinates?
5. How would the calculation be affected if the curve were defined implicitly?

### Tip
For curves with complicated expressions for \(\frac{dy}{dx}\), numerical integration can be a practical method to obtain accurate arc lengths.

Set-up the integral to calculate the length of the curve y = [(x^3) / 2] + [1 / (6x)] for 3 ≤ x ≤ 5. Evaluate the integral.

Learn how to set up and evaluate the integral to calculate the arc length of the curve y = (x^3 / 2) + (1 / 6x) over the interval 3 ≤ x ≤ 5. This solution involves the arc length formula and requires knowledge of integral calculus and derivatives. Suitable for advanced high school and college-level calculus students.

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