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I just started learning about parametric curves and I find it confusing that we have a 3rd variable but this 3rd variable "t" is some imaginary variable....I dont get what the difference is between this parameter "t" and the good ole 3 variable functions...This gets confusing for me especially when I do area in a bounded region integration related questions because I am trying to understand what this variable "t" for parameter really means...is this just some fake number or the 3rd dimension Z?

The example below uses t as the bounds of integration but why do that? Doesn't this graph look just like you regular integration graph where you can just use the x bounds for finding the area bounded by the curve?

enter image description here

NOTE: this example has an error in it, where it says t= 3/2 it should be t = 2

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In your example, you could actually do what you suggest - namely find $x$ bounds for the curve in question - but in order to do that you would have to rewrite the parametric curve in terms of $y$ and $x$ only:

$$x^2=t^2-2+t^{-2}\\ y^2=t^2+2+t^{-2}\\ \Rightarrow y^2=x^2+4.$$

From this point you could proceed as normal. We do not have to do this for any reason, however, and in fact since the curve is already parametrized it is just as easy to continue on in that way.

It is usually easiest to think of a parametrization by describing the values of $x$ and $y$ over a period of time. In the two dimensional case as you have, let's say that we have a fly walking around on our coordinate grid. At any particular time $t$, the fly has an $x$ value and a $y$ value, and we describe each as a function of time $t$.

We could also, as you note, imagine that $t=z$ in the Cartesian three dimensional space. This helpful way of thinking about a parametrized curve is no longer helpful when we consider a three dimensional case - imagine a fly buzzing through your room, and at any time it's $x$, $y$, and $z$ coordinate can be given as a function of $t$. Although it's true that this is the same idea as a four dimensional system, we can no longer visualize it usefully this way. We can, however, consider the path traced out over time in three dimensions, as if the fly were emitting a trail of dust which stayed in place in midair.

Without going too far off course, I would recommend thinking of the parameter $t$ as time. When integrating in two dimensions, imagine the fly walking along the paper, leaving a trail of it's path as it goes.

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omg that fly analogy is so cool im gonna be thinking about it on my exams :P –  Raynos Dec 17 '12 at 0:26
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I just started learning about parametric curves and I find it confusing that we have a 3rd variable but this 3rd variable "t" is some imaginary variable....I dont get what the difference is between this parameter "t" and the good ole 3 variable functions...

Hopefully the visualization below will help convey the different types of information that can be communicated by expressing a function in parametric form.

Here, we are taking the standard parametrization of the unit circle:

$$x(t)=\cos t, \quad y(t)=\sin t, \quad 0\le t\le 2\pi.$$

enter image description here

Top left: plot of the space curve $(t,x(t),y(t))$ in $\mathbb{R}^3$; $t$, $x$, and $y$ information is all available simultaneously

Top right: plot of the plane curve $(x(t),y(t))$; at time $t^*$, the point on the graph is $(t^*,x(t^*),y(t^*))$; $t$ information is suppressed in the static version (usually shown in a textbook) although we gain that information in an animation like this; plot conveys $x$-$y$ relationship over time; usually called the phase plane/space in applications

Bottom left: plot of the plane curve $t$ vs. $x(t)$; at time $t^*$, the point on the graph is $(t^*,x(t^*))$; conveys $t$-$x$ information; usually called time-state space in applications

Bottom right: plot of the plane curve $t$ vs $y(t)$; at time $t^*$, the point on the graph is $(t^*,y(t^*))$; conveys $t$-$y$ information; usually called time-state space in applications

PS Think of the black dot as the "fly" mentioned in the other answer. The curve traced out is the flight path of the fly.

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I am not much of a visual person when it comes to graphing so this just adds to the confusion.. –  Raynos Dec 17 '12 at 1:14
    
Well, if you're not visual when it comes to graphing, not much I can do to help... –  JohnD Dec 17 '12 at 1:18
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