Chapter 8.1, Problem 14E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Find the exact length of the curve.14. y = ln(cos x), 0 ≤ x ≤ π/3

To determine

The exact length of the curve.

Explanation

Given information:

The curve function is y=ln(cosx) (1)

The limits are a=0 and b=Ï€3.

Calculation:

The expression to find the length of the curve (L) is shown below:

L=âˆ«ab1+(dydx)2dx (2)

Here, the derivative of the function y is dydx, the lower limit is a, and the upper limit is b.

Differentiate Equation (1) with respect to x.

dydx=1cosx(âˆ’sinx)=âˆ’tanx

Substitute (âˆ’tanx) for dydx, 0 for a, and Ï€3 for b in Equation (2).

L=âˆ«0Ï€31+(âˆ’tanx)2dx=âˆ«0Ï€3

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