
Estimate the length of the curve f(x)= 1/x on 1 ≤ x ≤ 2. Using n=4


To estimate the length of the curve \( f(x) = \frac{1}{x} \) over the interval \( 1 \leq x \leq 2 \), we use the formula for arc length:

\[
L \approx \sum_{i=1}^{n} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \Delta x
\]

### Steps:
1. **Arc Length Formula:**  
   The arc length of a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by:

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

2. **Find the derivative \( \frac{dy}{dx} \):**  
   Since \( f(x) = \frac{1}{x} \), its derivative is:

   \[
   \frac{dy}{dx} = -\frac{1}{x^2}
   \]

3. **Square of the derivative:**  
   Now, square the derivative:

   \[
   \left( \frac{dy}{dx} \right)^2 = \frac{1}{x^4}
   \]

4. **Arc length expression:**  
   Substituting into the arc length formula:

   \[
   L = \int_{1}^{2} \sqrt{1 + \frac{1}{x^4}} \, dx
   \]

5. **Using numerical estimation with \( n = 4 \):**
   Divide the interval \( [1, 2] \) into 4 equal subintervals, each of width \( \Delta x = \frac{2 - 1}{4} = 0.25 \). The \( x \)-values at which we will evaluate are \( x_0 = 1 \), \( x_1 = 1.25 \), \( x_2 = 1.5 \), \( x_3 = 1.75 \), and \( x_4 = 2 \).

   We then compute the function \( \sqrt{1 + \frac{1}{x_i^4}} \) at each of these points.

Now I will compute this.The estimated length of the curve \( f(x) = \frac{1}{x} \) on the interval \( 1 \leq x \leq 2 \) using \( n = 4 \) subdivisions is approximately **1.187** units.

Would you like a more detailed explanation of this process or have any questions?

Here are five related questions to explore:
1. How would the result change if we increased \( n \)?
2. What is the exact arc length using calculus instead of estimation?
3. How do different functions affect the complexity of the arc length calculation?
4. Can we use Simpson’s rule for arc length estimation, and how would that differ?
5. What happens to the arc length if we change the interval, say from \( x = 1 \) to \( x = 3 \)?

**Tip:** Increasing the number of subdivisions \( n \) in numerical methods will generally improve accuracy, though it also increases computation time.

Estimate the length of the curve f(x) = 1/x on 1 ≤ x ≤ 2 using n=4.

Learn how to estimate the arc length of the curve f(x) = 1/x over the interval from x=1 to x=2 using numerical methods with n=4 subdivisions. This solution covers calculus concepts, including the arc length formula and numerical approximation techniques. Suitable for students in Grades 11-12.

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