How Do You Calculate the Area Inside a Parametric Loop?

[Sebobas]

Hey Guys!

Homework Statement

Find the area of the "loop" (I'm guessing it's called) formed by the set of parametrics:

--> x(t)=t³-3t and y(t)=t²+t+1

Homework Equations

I've already drawn the graph and as said before the curve creates a "loop", I have to find the area insed that "loop" and I don't know where to start.

The Attempt at a Solution

Here is the graph: http://www.wolframalpha.com/input/?i=Parametric+plot+(t^3-3t,t^2+t+1)

I know I probably have to integrate using y*dx..but I don't know where to separate the loop in order to integrate. I have no idea what the limits are.

Thanks in advance!

 

[SammyS]

Hey Guys!

Homework Statement

Find the area of the "loop" (I'm guessing it's called) formed by the set of parametrics:

--> x(t)=t³-3t and y(t)=t²+t+1

Homework Equations

I've already drawn the graph and as said before the curve creates a "loop", I have to find the area inside that "loop" and I don't know where to start.

The Attempt at a Solution

Here is the graph: http://www.wolframalpha.com/input/?i=Parametric+plot+(t^3-3t,t^2+t+1)

I know I probably have to integrate using y*dx..but I don't know where to separate the loop in order to integrate. I have no idea what the limits are.

Thanks in advance!

Hello Sebobas. Welcome to PF !

Suppose that you know y as a function of x. In fact there would be one function for the upper portion of the "loop", call it f₁(x), and another for the lower portion of the "loop", call it f₂(x).

Let x_L be the x value at the left-hand extreme of the "loop", and x_R be the x value at the left-hand extreme of the "loop".

Then the area that you want is, of course,

$\displaystyle \large {\int_{x_L}^{x_R}}{f_1(x)}\,dx-\large {\int_{x_L}^{x_R}}{f_2(x)}\,dx\ .$

Also notice that $\displaystyle \int_{x_L}^{x_R}{f_2(x)}\,dx=-\int_{x_R}^{x_L}{f_2(x)}\,dx\ .$

And of course, as in integration by substitution, $\displaystyle \int f(x)\,dx=\int g(t)\,\frac{dx}{dt}dt\,,$ where g(t)=f(x(t)) .

 

[LCKurtz]

Hey Guys!

Homework Statement

Find the area of the "loop" (I'm guessing it's called) formed by the set of parametrics:

--> x(t)=t³-3t and y(t)=t²+t+1



I know I probably have to integrate using y*dx..but I don't know where to separate the loop in order to integrate. I have no idea what the limits are.

Thanks in advance!

That graph looks like the x-y coordinates of the intersection point are $(-2,3)$. You know $t$ is between -2.497 and 2.497. You can figure out the two values of $t$ by inspection.

 

[Sebobas]

Hello Sebobas. Welcome to PF !

Suppose that you know y as a function of x. In fact there would be one function for the upper portion of the "loop", call it f₁(x), and another for the lower portion of the "loop", call it f₂(x).

Let x_L be the x value at the left-hand extreme of the "loop", and x_R be the x value at the left-hand extreme of the "loop".

Then the area that you want is, of course,

$\displaystyle \large {\int_{x_L}^{x_R}}{f_1(x)}\,dx-\large {\int_{x_L}^{x_R}}{f_2(x)}\,dx\ .$

Also notice that $\displaystyle \int_{x_L}^{x_R}{f_2(x)}\,dx=-\int_{x_R}^{x_L}{f_2(x)}\,dx\ .$

And of course, as in integration by substitution, $\displaystyle \int f(x)\,dx=\int g(t)\,\frac{dx}{dt}dt\,,$ where g(t)=f(x(t)) .

Hey Sammy! Thanks, glad to have found the community (:

I get what you mean, I even tried what you said, but I found myself with a problem..

...and that was that when trying to find y in function of x, t gave me 2 different values, which one do I use?

 

[Sebobas]

That graph looks like the x-y coordinates of the intersection point are $(-2,3)$. You know $t$ is between -2.497 and 2.497. You can figure out the two values of $t$ by inspection.

Hey Kurtz.. well you are right the graphs do intersect at (2,3), when t = 1 and t =-2 ...but what do I do then? Separate the curves? in that case what are the upper and lower limits of those new integrals?

 

[LCKurtz]

Hey Kurtz.. well you are right the graphs do intersect at (2,3), when t = 1 and t =-2 ...but what do I do then? Separate the curves? in that case what are the upper and lower limits of those new integrals?

Well, you mentioned in your original post about integrating ydx as a line integral. What do you get if you integrate$\int_C -ydx + 0dy$ counterclockwise around a loop? Think about Green's theorem. And you can do the line integral as a function of $t$.

 

[SammyS]

Hey Sammy! Thanks, glad to have found the community (:

I get what you mean, I even tried what you said, but I found myself with a problem..

...and that was that when trying to find y in function of x, t gave me 2 different values, which one do I use?

You don't need to use y as a function of x. I was just showing how one might think about the problem.

Basically the g(t) I mentioned in that last step gives y as a function of t: y = f_{(1 OR 2)}(x(t)) = t² + t + 1 .

dx/dt = 3t² - 3 .