https://file.aiquickdraw.com/mathbot/file/1726516189210ndm193d1.jpg

In the given image, we are tasked with finding the arc length \( s \) of a curve using the arc length formula:

\[
s = \int_{a}^{b} \sqrt{1 + (y')^2} \, dx
\]

### Steps to Solve:
1. **Substitute the value of \( y' \)**: From the image, \( y' = \frac{4}{5} \). Thus, the equation becomes:

\[
s = \int_{3}^{8} \sqrt{1 + \left(\frac{4}{5}\right)^2} \, dx
\]

2. **Simplify inside the square root**:
   \[
   \left(\frac{4}{5}\right)^2 = \frac{16}{25}
   \]
   So, the equation becomes:
   \[
   s = \int_{3}^{8} \sqrt{1 + \frac{16}{25}} \, dx
   \]

3. **Simplify further**:
   \[
   1 + \frac{16}{25} = \frac{25}{25} + \frac{16}{25} = \frac{41}{25}
   \]
   Therefore, the integral becomes:
   \[
   s = \int_{3}^{8} \sqrt{\frac{41}{25}} \, dx
   \]

4. **Simplify the square root**:
   \[
   \sqrt{\frac{41}{25}} = \frac{\sqrt{41}}{5}
   \]
   The arc length formula now simplifies to:
   \[
   s = \int_{3}^{8} \frac{\sqrt{41}}{5} \, dx
   \]

5. **Factor out constants**:
   Since \( \frac{\sqrt{41}}{5} \) is a constant, we can factor it out of the integral:
   \[
   s = \frac{\sqrt{41}}{5} \int_{3}^{8} 1 \, dx
   \]

6. **Evaluate the integral**:
   The integral \( \int_{3}^{8} 1 \, dx \) is simply the difference between the bounds, which is:
   \[
   8 - 3 = 5
   \]

7. **Final Calculation**:
   Now, multiply by the constant:
   \[
   s = \frac{\sqrt{41}}{5} \times 5 = \sqrt{41}
   \]

Thus, the arc length \( s \) is:

\[
s = \sqrt{41}
\]

### Final Answer:
\[
s = \sqrt{41}
\]

---

Would you like more details or have any other questions? Here are some related questions to explore further:

1. How does the arc length formula change if the curve is parametrized differently?
2. What are the conditions under which the arc length formula applies?
3. Can we derive the arc length for a parametric curve?
4. How do we find arc lengths of curves in polar coordinates?
5. How would you compute this arc length numerically if the integral were more complex?

**Tip**: Always check for constants and simplifications before performing an integral—it can make the calculation much easier.

Substitute the values of y' in the formula for arc length and solve for the distance s.

Learn how to solve an arc length problem using the definite integral formula s = ∫[3, 8] √(1 + (4/5)²) dx. This step-by-step solution simplifies the square root and evaluates the integral.

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