find the arc length of the curve y=x-lnx on interval [1,4]

To find the arc length of the curve \( y = x - \ln{x} \) on the interval \( [1, 4] \), we use the formula for the arc length of a curve:

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

### Step 1: Find the derivative of \( y \)
Given \( y = x - \ln{x} \), we first find \( \frac{dy}{dx} \):

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

### Step 2: Square the derivative
Now, square the derivative:

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

### Step 3: Plug into the arc length formula
Substitute this into the arc length formula:

\[
L = \int_1^4 \sqrt{1 + 1 - 2\frac{1}{x} + \frac{1}{x^2}} \, dx
\]

Simplify the expression under the square root:

\[
L = \int_1^4 \sqrt{2 - 2\frac{1}{x} + \frac{1}{x^2}} \, dx
\]

### Step 4: Simplify further
Factor out a 2:

\[
L = \int_1^4 \sqrt{\frac{(x - 1)^2}{x^2}} \, dx
\]

This simplifies to:

\[
L = \int_1^4 \frac{|x - 1|}{x} \, dx
\]

Since \( x \geq 1 \) in the interval \( [1, 4] \), we can drop the absolute value:

\[
L = \int_1^4 \frac{x - 1}{x} \, dx = \int_1^4 \left( 1 - \frac{1}{x} \right) \, dx
\]

### Step 5: Compute the integral
Now, integrate each term:

\[
L = \int_1^4 1 \, dx - \int_1^4 \frac{1}{x} \, dx
\]

\[
L = \left[ x \right]_1^4 - \left[ \ln{x} \right]_1^4
\]

\[
L = (4 - 1) - (\ln{4} - \ln{1})
\]

\[
L = 3 - \ln{4}
\]

Since \( \ln{1} = 0 \), the final arc length is:

\[
L = 3 - \ln{4}
\]

Thus, the arc length of the curve is approximately \( L \approx 3 - 1.386 = 1.614 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the derivative of logarithmic functions?
2. Can the arc length formula be applied to parametric curves?
3. How would the result change if the interval was different, say [1, 5]?
4. What happens if the curve's equation changes to \( y = x + \ln{x} \)?
5. Can arc length be calculated using numerical methods for complex curves?

**Tip**: When simplifying expressions inside a square root, look for opportunities to factor or simplify terms before integrating to avoid complex computations.

Learn how to calculate the arc length of the curve y = x - ln(x) on the interval [1, 4]. This detailed solution covers calculus concepts, including finding the derivative, simplifying the arc length formula, and computing the integral. Ideal for undergraduate calculus students.

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