*y* = 8*x*3⁄2 + 7 how to find arc length

To find the arc length of a function \( y = 8x^{3/2} + 7 \), we can use the formula for arc length of a curve between two points \( x = a \) and \( x = b \):

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

### Steps:

1. **Differentiate the function:**
   First, find the derivative of the given function \( y = 8x^{3/2} + 7 \).

   \[
   \frac{dy}{dx} = \frac{d}{dx}(8x^{3/2}) = 8 \cdot \frac{3}{2}x^{1/2} = 12x^{1/2}
   \]

2. **Square the derivative:**

   \[
   \left(\frac{dy}{dx}\right)^2 = (12x^{1/2})^2 = 144x
   \]

3. **Substitute into the arc length formula:**

   \[
   L = \int_a^b \sqrt{1 + 144x} \, dx
   \]

4. **Solve the integral:**

   Use a substitution to solve the integral \( \int \sqrt{1 + 144x} \, dx \). Let \( u = 1 + 144x \), so that \( du = 144 \, dx \), and then proceed to compute the integral.

5. **Evaluate the integral:**

   Once you have the integral, evaluate it from \( x = a \) to \( x = b \), depending on the specific bounds you are given or interested in.

Do you have specific limits for \( a \) and \( b \) to evaluate this arc length? Would you like further details on solving the integral?

### Related Questions:
1. How do you differentiate functions with fractional exponents?
2. What is the geometric interpretation of arc length in calculus?
3. Can you find arc length for parametric curves?
4. What other methods are used to approximate arc lengths of curves?
5. How is the concept of arc length applied in physics?

### Tip: 
When dealing with integrals involving square roots, consider substitutions or trigonometric identities to simplify the process!

Learn how to find the arc length of the function y = 8x^(3/2) + 7 using calculus. This detailed guide covers key steps, including finding derivatives, squaring them, and solving the arc length integral. Suitable for advanced high school and college calculus students.

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