Find the length of the curve yequalsln left parenthesis cosine x right parenthesis​, 0less than or equalsxless than or equalsStartFraction pi Over 4 EndFraction
.

To find the length of the curve \( y = \ln(\cos(x)) \) for \( 0 \leq x \leq \frac{\pi}{4} \), 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: Compute \(\frac{dy}{dx}\)

Given \( y = \ln(\cos(x)) \), we apply the chain rule to differentiate:

\[
\frac{dy}{dx} = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x)
\]

### Step 2: Set up the arc length formula

The arc length formula becomes:

\[
L = \int_0^{\frac{\pi}{4}} \sqrt{1 + (-\tan(x))^2} \, dx
\]

Since \( \tan^2(x) + 1 = \sec^2(x) \), we simplify the integrand:

\[
L = \int_0^{\frac{\pi}{4}} \sqrt{\sec^2(x)} \, dx = \int_0^{\frac{\pi}{4}} \sec(x) \, dx
\]

### Step 3: Integrate

The integral of \( \sec(x) \) is known:

\[
\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)|
\]

Thus, the arc length is:

\[
L = \left[ \ln|\sec(x) + \tan(x)| \right]_0^{\frac{\pi}{4}}
\]

### Step 4: Evaluate the bounds

At \( x = \frac{\pi}{4} \):

\[
\sec\left(\frac{\pi}{4}\right) = \sqrt{2}, \quad \tan\left(\frac{\pi}{4}\right) = 1
\]

So,

\[
\ln|\sec\left(\frac{\pi}{4}\right) + \tan\left(\frac{\pi}{4}\right)| = \ln(\sqrt{2} + 1)
\]

At \( x = 0 \):

\[
\sec(0) = 1, \quad \tan(0) = 0
\]

So,

\[
\ln|\sec(0) + \tan(0)| = \ln(1) = 0
\]

### Step 5: Final answer

The length of the curve is:

\[
L = \ln(\sqrt{2} + 1)
\]

This is the exact length of the curve.

---

Would you like more details or have any further questions? Here are 5 related questions to explore further:

1. What is the geometric interpretation of the curve \( y = \ln(\cos(x)) \)?
2. How do we derive the formula for arc length in calculus?
3. What are some real-world applications of curve length calculations?
4. How does the chain rule work when differentiating logarithmic functions?
5. Can we approximate the length of a curve using numerical methods?

**Tip:** In problems involving arc length, always remember to simplify the square root expressions as much as possible before integrating!

Find the length of the curve y = ln(cos(x)), 0 ≤ x ≤ π/4.

Learn how to calculate the arc length of the curve y = ln(cos(x)) for 0 ≤ x ≤ π/4. This problem involves differentiation, arc length formula, and trigonometric identities. Suitable for students in Grades 11-12 or early college calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation