Chapter 10.5, Problem 57E

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Finding the Arc Length of a Polar Curve In Exercises 53-58, find the length of the curve over the given interval. r = 1 + sin θ ,       [ 0 , 2 π ]

To determine

To calculate: The length of curve the polar equation r=1+sinθ in the interval [0,2π].

Explanation

Given:

The polar equation is r=1+sinÎ¸ and the interval is [0,2Ï€].

Formula Used:

Length of curve of polar coordinates

s=âˆ«Î²Î±r2+(drdÎ¸)2dÎ¸

Here, s is arc length of polar region is, r is distance from the origin of the curve. Î± And Î² are the intervals and r=f(Î¸).

d(a)dx=0, d(sinx)dx=cosx and sin2Î¸+cos2Î¸=1 here â€˜aâ€™ is a constant

Calculation:

The length of curve of polar coordinates is:

s=âˆ«Î²Î±r2+(drdÎ¸)2dÎ¸

For drdÎ¸,

drdÎ¸=ddÎ¸(1+sinÎ¸)drdÎ¸=cosÎ¸

So apply the formula and put for r=1+sinÎ¸, drdÎ¸=cosÎ¸ Î±=2Ï€ and Î²=0

s=âˆ«02Ï€(1+sinÎ¸)2+(<

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