Find the length of the following curve.
y equals 3 x Superscript three halves
 from xequals0 to xequals8

To find the length of the curve \( y = 3x^{\frac{3}{2}} \) from \( x = 0 \) to \( x = 8 \), we use the arc length formula for a function \( y = f(x) \):

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

### Step 1: Find \( \frac{dy}{dx} \)
Given \( y = 3x^{\frac{3}{2}} \), we first compute the derivative \( \frac{dy}{dx} \).

\[
\frac{dy}{dx} = 3 \cdot \frac{3}{2} \cdot x^{\frac{1}{2}} = \frac{9}{2} \cdot x^{\frac{1}{2}}
\]

### Step 2: Set up the arc length formula
Now, substitute \( \frac{dy}{dx} \) into the arc length formula:

\[
L = \int_{0}^{8} \sqrt{1 + \left(\frac{9}{2} \cdot x^{\frac{1}{2}}\right)^2} \, dx
\]

Simplify the expression inside the square root:

\[
L = \int_{0}^{8} \sqrt{1 + \frac{81}{4}x} \, dx
\]

### Step 3: Solve the integral
We can now evaluate the integral:

\[
L = \int_{0}^{8} \sqrt{1 + \frac{81}{4}x} \, dx
\]

This can be solved using substitution. Let \( u = 1 + \frac{81}{4}x \), then compute the integral accordingly. I’ll now solve this integral.

Let me calculate it.The length of the curve \( y = 3x^{\frac{3}{2}} \) from \( x = 0 \) to \( x = 8 \) is approximately \( 68.48 \) units.

Would you like more details on any part of the process?

Here are some related questions to explore:
1. How does the arc length formula change if the curve is parameterized differently?
2. What if the function was \( y = 3x^2 \)? How would the arc length change?
3. Can you use the arc length formula for parametric curves?
4. How would the arc length calculation differ if \( y = 3x^{1/2} \) instead?
5. What are some real-life applications of arc length calculations?

**Tip**: For curves expressed in parametric form, the arc length formula adjusts to incorporate both the \( x \)- and \( y \)-derivatives with respect to a parameter.

Find the length of the following curve. y equals 3x Superscript three halves from x equals 0 to x equals 8.

Learn how to find the length of the curve y = 3x^(3/2) from x = 0 to x = 8 using the arc length formula. This problem involves calculus concepts like derivatives and integration, making it suitable for students in Grades 11-12.

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