   Chapter 8, Problem 2RE

Chapter
Section
Textbook Problem

Find the length of the curve.2. y = 2 ln ( sin 1 2 x ) , π/3 ≤ x ≤ π

To determine

The length of the curve.

Explanation

Given information:

The curve function is y=2ln(sin12x) (1)

The limits are a=π3 and b=π .

Calculation:

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

L=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=21sin12x(cos12x)(12)=cos12xsin12x=cot12x

Substitute cot12x for dydx , π3 for a, and π for b in Equation (2).

L=π3π1+(cot2(12x))dx=π3πcsc2(12x)dx=π3πcsc(12x)dx (3)

Let u=12x (4)

Differentiate Equation (4) with respect to x

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