Turn on thread page Beta

help! intrinsic coordinates watch

Announcements
    • Thread Starter
    Offline

    0
    ReputationRep:
    ive found the book to be completely useless when explaining these... can anyone help by solving these questions?

    3. Measuring arc length from the origin O, find the intrinsic equation of the curve given parametically by x=2t+2sint and y=2-2cost. [ans: s=8sin(psi)]

    6. For the curve with equation siny=e^x, where s is measured from the point (0, pi/2), show that e^s=tan(psi/2).

    thanks!
    Offline

    2
    ReputationRep:
    I think it in P5 book:

    The solution might be:

    dx/dt = 2 + 2cos t
    dy/dt = 2sin t

    the arc length is:

    s = INT ( sqrt [ (dx/dt)^2 + (dy/dt)^2 ] ) dt
    = INT ( sqrt [ 4*( 1+ cos t)^2 + 4*sin ^2 t ]) dt
    = INT ( 2* sqrt ( 2 + 2 cos t) ) dt
    = 2*[ 2/3*( 2 + 2 cos t)^ 3/2 * ( 2t + 2sin t) ]

    I don't know which is the limit of that? From 0 to Pi or what?
    Can you tell me cearly?
    Offline

    0
    ReputationRep:
    (Original post by VCVT17)
    = INT ( 2* sqrt ( 2 + 2 cos t) ) dt
    = 2*[ 2/3*( 2 + 2 cos t)^ 3/2 * ( 2t + 2sin t) ]
    i've never been taught to integrate like this...could you please share how; you came to this?

    cheers

    PK
    Offline

    2
    ReputationRep:
    Integrating by substituting:
    Let sqrt ( 2 + 2 cos t) = a ^ 1/2
    So integrating it become: 2/3*a^1/2
    And then integrate a as it has 2t+ 2sin t

    I will check again...
    Offline

    2
    ReputationRep:
    Sorry I did wrong

    It must be:
    = INT ( 2* sqrt ( 2 + 2 cos t) ) dt
    = 2*INT(sqrt (4* cos ^2 t/2) ) dt
    = 4* INT( cos t/2) dt
    = 8* sin t/2

    Sorry about that
    Offline

    0
    ReputationRep:
    = INT ( 2* sqrt ( 2 + 2 cos t) ) dt
    = 2*[ 2/3*( 2 + 2 cos t)^ 3/2 * ( 2t + 2sin t) ]

    still,

    how do you integrate s function of a function like above. I mean, i've been taught how to differentaite by teh chain rule. Is that a chain fule of integration or something?...bit lost...please enlighten me as to whether you can actually integrate stuff using stuff like what you did above?

    thanks

    PK
    Offline

    10
    ReputationRep:
    3.
    dx/dt = 2(1+cost)
    dy/dt = 2sint
    dy/dx = sint/(1+cost) = 2sin(t/2)cos(t/2)/(1+2cos²(t/2)-1) = sin(t/2)/cos(t/2) = tan(t/2) = tan(psi) => psi = (t/2)

    (dx/dt)² + (dy/dt)² = 4+8cost+4cos²t + 4sin²t = 8(1+cost)
    sqrt[(dx/dt)² + (dy/dt)²] = sqrt[8(1+cost)] = sqrt[8(1+2cos²(t/2)-1)] = 4cos(t/2)

    s = ∫ sqrt[(dx/dt)² + (dy/dt)²] dt = ∫ 4cos(t/2) dt = 8sin(t/2) = 8sin(psi)

    6.
    siny = e^x => x = ln(siny)
    dx/dy = coty = cot(psi) => psi = y

    s = ∫ sqrt[1+(dx/dy)²] dy = ∫ sqrt[1+cot²y] dy = ∫ cosecy dy = ln|tan(y/2)| = ln|tan(psi/2)|
    e^s = tan(psi/2)
    Offline

    10
    ReputationRep:
    (Original post by Phil23)
    = INT ( 2* sqrt ( 2 + 2 cos t) ) dt
    = 2*[ 2/3*( 2 + 2 cos t)^ 3/2 * ( 2t + 2sin t) ]

    still,

    how do you integrate s function of a function like above. I mean, i've been taught how to differentaite by teh chain rule. Is that a chain fule of integration or something?...bit lost...please enlighten me as to whether you can actually integrate stuff using stuff like what you did above?

    thanks

    PK
    That's wrong. I think he applied the 'rule' for integrating (ax+b)^n to (2+2cost)^(0.5), but you can't do that since 2+2cost isn't linear.
    Offline

    0
    ReputationRep:
    (Original post by dvs)
    That's wrong. I think he applied the 'rule' for integrating (ax+b)^n to (2+2cost)^(0.5), but you can't do that since 2+2cost isn't linear.
    thanks for that...thought that method he employed didn't ring any bells
 
 
 
Turn on thread page Beta
Updated: March 4, 2005

University open days

  • University of the West of England, Bristol
    Undergraduate Open Afternoon - Frenchay Campus Undergraduate
    Wed, 23 Jan '19
  • University of East London
    Postgraduate Open Evening Postgraduate
    Wed, 23 Jan '19
  • University of Gloucestershire
    School of Education Open Day Postgraduate
    Wed, 23 Jan '19
Poll
Brexit: Given the chance now, would you vote leave or remain?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.