12.3.2 Use the polar equation of the lemniscate and the formula for the element of arc in polar coordinates, V(rde)2 + dr2 ds To deduce that the arc length of the lemniscate is given by do S= r ds (rde)2 + dr2 (dx)2 + (dy)2 ds And r2 cos 20 = r4 = cos2 20 ds der2 + .d0 Consider r2 = cos 20 dr Here = 2 de Differentiating on both sides: sin 20 2 2r dr 2 sin 20 de ds = d0 r2 sin 20 de dr= sin2 20 ds de r4 + r2 dr sin 20 de de r4sin2 20 ds = de Vcos2 20sin2 20 ds de ds = Integrating on both sides: de ds = r S= 12.3.3 Conclude by changing the variable of integration to r, that the total length of the lemniscate is 4dr/V1 - r4 Unlike the arcsine integrand 1/V1 - t2 which is rationalized by substituting 2v/(1+ v2) for t, the lemniscate integrand 1/V1 - t4cannot be rationalized by replacing t by any rational function.
12.3.2 Use the polar equation of the lemniscate and the formula for the element of arc in polar coordinates, V(rde)2 + dr2 ds To deduce that the arc length of the lemniscate is given by do S= r ds (rde)2 + dr2 (dx)2 + (dy)2 ds And r2 cos 20 = r4 = cos2 20 ds der2 + .d0 Consider r2 = cos 20 dr Here = 2 de Differentiating on both sides: sin 20 2 2r dr 2 sin 20 de ds = d0 r2 sin 20 de dr= sin2 20 ds de r4 + r2 dr sin 20 de de r4sin2 20 ds = de Vcos2 20sin2 20 ds de ds = Integrating on both sides: de ds = r S= 12.3.3 Conclude by changing the variable of integration to r, that the total length of the lemniscate is 4dr/V1 - r4 Unlike the arcsine integrand 1/V1 - t2 which is rationalized by substituting 2v/(1+ v2) for t, the lemniscate integrand 1/V1 - t4cannot be rationalized by replacing t by any rational function.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 80E
Related questions
Concept explainers
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Topic Video
Question
100%
I need help with 12.3.3.please.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning