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Parametric Intergration question

  • Thread starter thomas49th
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  • #1
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Homework Statement



a curve has parametric equations:
x = t - 2sin t
y = 1 - 2cos t


R is enclosed by the curve and the x axis. Show that the area of R is given by the integral:

[tex]\int^{\frac{5\pi}{3}}_{\frac{\pi}{3}} (1-2\cos t)^{2}[/tex]

Homework Equations





The Attempt at a Solution



Not sure what to do :\
 

Answers and Replies

  • #2
CompuChip
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Suppose that the curve was given in the form y = f(x), then how would you calculate the area? Now do the variable substitution from x to t which is given.
 
  • #3
Defennder
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You might want to make use of this:
[tex]\int f(x) dx = \int f(t) \frac{dx}{dt} \ dt[/tex]
By the way what's this mathematical property called?
 
  • #4
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ermm mathematical propety???

dx/dt = 1 - 2cost


i have to multiply this by f(t) (how do i found that) and then intergrate that with respect to t?

Thanks
 
  • #5
Defennder
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Yes, the integration is done wrt t. Isn't that given in the answer? And as for f(t), you should know that f(x) in the original integral f(x)dx represents a particular variable. That variable must now be expressed in terms of t for the integration to work.
 
  • #6
tiny-tim
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[tex]\int f(x) dx = \int f(t) \frac{dx}{dt} \ dt[/tex]
Hi thomas49th! :smile:

I suspect you're confused by the (x) and the (t).

Just write it [tex]\int f dx = \int f \frac{dx}{dt} \ dt[/tex]

or, in this case, [tex]\int y dx = \int y \frac{dx}{dt} \ dt[/tex] :smile:
 
  • #7
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ahh yeh it's easy

got it!

cheers :)
 
  • #8
CompuChip
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The property is called substitution of variables. Probably you've seen it in this form:
Solve [tex]\int \sin(t) \cos(t) \, dt[/tex]; let [itex]u = \sin(t)[/itex], then [itex]du = \cos(t) \, dt[/itex] so [tex]\int \sin(t) \cos(t) \, dt = \int u \, du = \frac12 u^2 + C = \frac12 \sin^2(t) + C[/tex].
You'll notice that in switching from the variable t to the variable u, I replaced [itex]\frac{du}{dt} dt[/itex] by [itex]du[/itex].

If you have never seen this, forget it. Otherwise I hope it clarifies what just happened :)
 

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