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. Let the parametric representation for the ellipse ( x 5 ) 2 + ( y 3 ) 2 = 1 be x = 5 cos t ; y = 3 sin t ; 0 ≤ t ≤ 2 π Then differentiating both sides of above representation with respect to t , we have that x ′ ( t ) = − 5 sin t y ′ ( t ) = 3 cos t Now, the arc length is evaluated as s ( t ) = ∫ 0 2 π [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 d t = ∫ 0 2 π ( − 5 sin t ) 2 + ( 3 cos t ) 2 d t = ∫ 0 2 π 9 + 16 sin 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 ) = 3.244667 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 ( 3.244667 + 4.412724 + .... + 3.244667 ) = 25.528308 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 , ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Chapter 12.2, Problem 27E

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To plot: the plot the ellipse ${(\frac{x}{5})}^{2} + {(\frac{y}{3})}^{2} = 1$ and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

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Chapter 12.2, Problem 26E

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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 ellipse ( x 5 ) 2 + ( y 3 ) 2 = 1 and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

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

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To plot: the plot the ellipse ${(\frac{x}{5})}^{2} + {(\frac{y}{3})}^{2} = 1$ 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 ellipse ( x 5 ) 2 + ( y 3 ) 2 = 1 and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length

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

Concept explainers

Videos

To plot: the plot the ellipse (x5)2+(y3)2=1 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 ellipse ${(\frac{x}{5})}^{2} + {(\frac{y}{3})}^{2} = 1$ and use the midpoint rule with N=10, 20, 30 and 50 to approximate its length