Explanation For the vector function r ( t ) = ( f ( t ) g ( t ) h ( t ) ) In the interval a ≤ t ≤ b and x = f ( t ) y = g ( t ) z = h ( t ) Then length is given by L = ∫ ( f ' ( t ) ) 2 + ( g ' ( t ) ) 2 + ( h ' ( t ) ) 2 d t Now we know that ∥ r ' ( t ) ∥ = ( f ' ( t ) ) 2 + ( g ' ( t ) ) 2 + ( h ' ( t ) ) 2 ………………………………..(I) L = ∫ a b ∥ r ' ( t ) ∥ d t ……………………………………..(II) Now for r ( t ) = ( 2 t 2 + 1 , 2 t − 1 , t 3 ) 0 ≤ t ≤ 2 Then length is given by L = ∫ ( f ' ( t ) ) 2 + ( g ' ( t ) )

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic…

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along…

Triple Integral

Examples:

Videos

Question

Chapter 14.3, Problem 6E

To determine

The length of arc $r (t) = (2 t^{2} + 1, 2 t - 1, t^{3})$ $0 \leq t \leq 2$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.3, Problem 5E

Chapter 14.3, Problem 7E

Students have asked these similar questions

In an exercise we have to show that T: P(R) -> P(R) by T(p)(x) = p(x) - p(0): a) T is linear b) Is T one-to-one? c) Is T onto? I tried to sovle this exercise but I'm not sure whether it's correct.

Find the lengths of the curves in Exercises 19–22. 19. y = x^(1/2) - (1/3)x^(3/2), 1<= x <=4 20. x = y^(2/3), 1<=y<= 8 21. y = x2 - (ln x)/8, 1<=x <=2 22. x = (y3/12) + (1/y), 1<= y<=2.

Evaluate∮C (x + 3y)dx + ydy where C is the Jordan curve given by thegraphs of y = e^x, y = e^−x and the horizontal line y = e^−1a) By Green’s theoremb) By direct computation

Chapter 14 Solutions

CALCULUS (CLOTH)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The length of arc r ( t ) = ( 2 t 2 + 1 , 2 t − 1 , t 3 ) 0 ≤ t ≤ 2

Solution Summary: The author explains that the length of arc is L = 1 27 [ ( 68 ) 3 2 ( 32 ).

Concept explainers

Videos

The length of arc $r (t) = (2 t^{2} + 1, 2 t - 1, t^{3})$ $0 \leq t \leq 2$

Want to see the full answer?

Chapter 14 Solutions

The length of arc r ( t ) = ( 2 t 2 + 1 , 2 t − 1 , t 3 ) 0 ≤ t ≤ 2

Solution Summary: The author explains that the length of arc is L = 1 27 [ ( 68 ) 3 2 ( 32 ).

Concept explainers

Videos

The length of arc r(t)=(2t2+1,2t−1,t3) 0≤t≤2

Want to see the full answer?

Chapter 14 Solutions

The length of arc $r (t) = (2 t^{2} + 1, 2 t - 1, t^{3})$ $0 \leq t \leq 2$