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Integrating parametric equations

  1. May 12, 2015 #1
    1. The problem statement, all variables and given/known data

    Why does
    [itex]
    \int_a^b \, y \; dx
    [/itex]
    become
    [itex]
    \int_\alpha^\beta \, g(t) f^\prime(t) \; dt
    [/itex]
    if x = f(t) and y = g(t) and alpha <= t <= beta?
    2. Relevant equations

    Substitution rule?


    3. The attempt at a solution

    I'm not sure how y = y(x) in the integrand turns into g(t). Isn't y a function of x in the first expression? How do they go from y(x) to g(t)?
     
  2. jcsd
  3. May 12, 2015 #2

    Mark44

    Staff: Mentor

    Substitution.
    Replace y by g(t). What should you replace dx by?
     
  4. May 12, 2015 #3
    If x = f(t), dx = f'(t) dt. I understand that part.

    But in
    [tex]
    \int y \; dx
    [/tex]

    isn't y = y(x) a function of x? We'd then have y = y(x) = y[x(t)]. How can we just let y = g(t) and get the resulting expression in t?
     
  5. May 12, 2015 #4

    Mark44

    Staff: Mentor

    No, not according to the problem description you wrote, which says y = g(t). x is a different function of t.
    Because it is given that y = g(t).
     
  6. May 12, 2015 #5

    Zondrina

    User Avatar
    Homework Helper

    You want to show:

    $$\int_a^b y \space dx = \int_\alpha^\beta g(t) f^\prime(t) \space dt$$

    When given:

    $$x = f(t)$$
    $$y = g(t)$$

    Write ##\frac{dx}{dt} = f^\prime(t)##; You also know ##y##.
     
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