Explanation Formula Used: Midpoint rule: The Nth midpoint approximation to ∫ a b f ( x ) d x is M N = Δ x ( f ( c 1 ) + f ( c 2 ) + .... f ( c N ) ) where Δ x = b − a N and c j = a + ( j − 1 2 ) Δ x is the midpoint of [ x j − 1 , x j ] Calculations: Graph of the given curve is shown below Now, we have to find the arc length using the midpoint rulewith N=10, 20, 30 and 50. Consider the curve c ( t ) = ( t − sin 2 t , 1 − cos 2 t ) for 0 ≤ t ≤ 2 π Then, x ( t ) = t − sin 2 t y ( t ) = 1 − cos 2 t So, x ′ ( t ) = 1 − 2 cos 2 t y ′ ( t ) = 2 sin 2 t Now, the arc length is evaluated as s ( t ) = ∫ 0 2 π [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 d t = ∫ 0 2 π ( 1 − 2 cos 2 t ) 2 + ( 2 sin 2 t ) 2 d t = ∫ 0 2 π 5 − 4 cos 2 t d t Now, first take N = 10 Here we have that a = 0 , b = 2 π , N = 10 Therefore, Δ t = 2 π − 0 10 = π 5 Now, divide the interval [ 0 , 2 π ] into N = 10 subintervals of length Δ t = π 5 with the following end points: a = 0 , π 5 , 2 π 5 , ... , 10 π 5 = 2 π = b Now, we just have to evaluate the function at the midpoints of the above sub-intervals: f ( t 0 + t 1 2 ) = f ( 0 + π 5 2 ) = f ( π 10 ) = 1.328131 Similarly, find the value of function at other points and finally just sum up these values and multiply by Δ t M 10 = π 5 ( 1.3281331 + 2.497212 + .... + 1.328131 ) = 13.384046 Now, take N = 20 Here we have that a = 0 , b = 2 π , N = 20 Therefore, Δ t = 2 π − 0 20 = π 10 Now, divide the interval [ 0 , 2 π ] into N = 20 subintervals of length Δ t = π 10 with the following end points: a = 0 , π 10 , 2 π 10 , ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 12.2, Problem 26E

To determine

To plot: the plot the curve $c (t) = (t - \sin 2 t, 1 - \cos 2 t) for 0 \leq t \leq 2 π$ and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

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Chapter 12.2, Problem 25E

Chapter 12.2, Problem 27E

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Chapter 12 Solutions

Calculus - Standalone book

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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To plot: the plot the curve c ( t ) = ( t − sin 2 t , 1 − cos 2 t ) for 0 ≤ t ≤ 2 π and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

Solution Summary: The author calculates the arc length using the midpoint rule with N=10, 20, 30 and 50.

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To plot: the plot the curve $c (t) = (t - \sin 2 t, 1 - \cos 2 t) for 0 \leq t \leq 2 π$ and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

Want to see the full answer?

Chapter 12 Solutions

To plot: the plot the curve c ( t ) = ( t − sin 2 t , 1 − cos 2 t ) for 0 ≤ t ≤ 2 π and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

Solution Summary: The author calculates the arc length using the midpoint rule with N=10, 20, 30 and 50.

Concept explainers

Videos

To plot: the plot the curve c(t)=(t−sin2t,1−cos2t) for 0≤t≤2π and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

Want to see the full answer?

Chapter 12 Solutions

To plot: the plot the curve $c (t) = (t - \sin 2 t, 1 - \cos 2 t) for 0 \leq t \leq 2 π$ and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length