Find the arc length of the graph of the function over the indicated interval. (Round your answer to three decimal places.) y = (3/2) times x^(2/3) + 9, [8, 64]

To find the arc length of a function \( y = f(x) \) over an interval \([a, b]\), we use the following formula:

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

Given the function:
\[
y = \frac{3}{2}x^{\frac{2}{3}} + 9
\]
and the interval \([8, 64]\), we first need to compute the derivative \(\frac{dy}{dx}\).

### Step 1: Compute the derivative
\[
y = \frac{3}{2}x^{\frac{2}{3}} + 9
\]

The derivative of \( y \) with respect to \( x \) is:
\[
\frac{dy}{dx} = \frac{d}{dx} \left(\frac{3}{2}x^{\frac{2}{3}} \right) = \frac{3}{2} \times \frac{2}{3} x^{\frac{2}{3} - 1} = x^{-\frac{1}{3}}
\]

Thus, \(\frac{dy}{dx} = x^{-\frac{1}{3}}\).

### Step 2: Set up the arc length integral
Now, substitute the derivative into the arc length formula:
\[
L = \int_8^{64} \sqrt{1 + \left(x^{-\frac{1}{3}}\right)^2} \, dx = \int_8^{64} \sqrt{1 + x^{-\frac{2}{3}}} \, dx
\]

### Step 3: Solve the integral
We'll compute this integral to find the arc length.

Let me calculate it.The arc length of the graph of the function \( y = \frac{3}{2}x^{\frac{2}{3}} + 9 \) over the interval \([8, 64]\) is approximately \( 58.900 \) units.

Do you want any further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does the derivative of a function affect the arc length?
2. What are the conditions under which the arc length formula can be applied?
3. Can arc length be calculated for functions with discontinuities?
4. How does changing the interval affect the arc length of the function?
5. What is the arc length of a straight line, and how does it compare to curves?

**Tip:** When calculating arc lengths, derivatives with larger values significantly impact the length, as they increase the slope of the curve.

Find the arc length of the function y = (3/2)x^(2/3) + 9 over the interval [8, 64]. Learn the steps involved, including derivative calculation and integration for the arc length formula. Suitable for calculus students in Grades 11-12 and college.

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