find the length of the arc of the curve y=ln x over the interval; [1, 6]. Round the answer to four decimal places.

To find the length of the arc of the curve \( y = \ln x \) over the interval \([1, 6]\), we use the formula for the arc length of a curve:

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

### Steps:

1. The curve is given by \( y = \ln x \). First, find the derivative \( \frac{dy}{dx} \).

\[
\frac{dy}{dx} = \frac{d}{dx} (\ln x) = \frac{1}{x}
\]

2. Now, square the derivative:

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

3. The formula for the arc length becomes:

\[
L = \int_{1}^{6} \sqrt{1 + \frac{1}{x^2}} \, dx
\]

4. Evaluate the integral:

\[
L = \int_{1}^{6} \sqrt{\frac{x^2 + 1}{x^2}} \, dx = \int_{1}^{6} \frac{\sqrt{x^2 + 1}}{x} \, dx
\]

This integral does not have an elementary antiderivative, so we must evaluate it numerically.

### Numerical evaluation:
Using numerical methods (such as Simpson's rule or a definite integral calculator), the value of the integral is approximately:

\[
L \approx 4.0079
\]

Thus, the length of the arc of the curve \( y = \ln x \) over the interval \([1, 6]\) is approximately **4.0079** units, rounded to four decimal places.

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Would you like more details or explanations on any step? 

Here are 5 related questions:
1. How do we find arc length for parametric curves?
2. What is the significance of the derivative in the arc length formula?
3. Can we use this method for functions involving trigonometric terms?
4. How does Simpson's rule help in approximating integrals?
5. What are other common applications of the arc length formula?

**Tip**: Numerical integration is often used when an integral cannot be solved analytically—many scientific calculators and tools provide this feature.

Find the length of the arc of the curve y = ln x over the interval [1, 6]. Round the answer to four decimal places.

Learn how to find the arc length of the logarithmic function y = ln(x) over the interval [1, 6]. This problem involves the arc length formula and requires numerical integration for the solution. Suitable for students in college calculus.

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