How do you find the parameters for line integrals in different shapes?

[sebastianzx6r]

How do you find the parameters for x,y,z and so forth. The examples in the book always use x=cos t and y=sint, but I know that there are more options. I'm just lost as to how to look at it.

For example

the line integral xy^4 ds, C is the right half of the circle x^2+y^2=16

I know this that eq. is a circle with origin as center and radius 4. So when it ways right half circle does that mean the parts of the circle in quadrants I and IV? And what parameters would you use and how would you know? What do you ask yourselves when you are working a problem like this?

Thanks

 

[Cincinnatus]

There isn't really any general method to parametrize any curve. Whenever you are working with a circle (or section of one) using x=cos(t), y=sin(t) for some values of t is the way to go.

In your example, you would parametrize the right half of that circle as:
x=4cos(t)
y=4sin(t)
where -pi/2<t<pi/2

 

[Office_Shredder]

To parametrize a curve, you just need to develop intuition. Mostly you'll be looking for trigonometric relationships (for closed curves), and polynomial relations. for open curves

x=acost
y=bsint

Is a general elliptical curve with axes of a and b. This is probably the most often used one

 

[HallsofIvy]

If it is possible to write y as a function of x: y= f(x) then you can use x itself as "parameter". More formally, x= t, y= f(t).

There are, if fact, an infinite number of different ways to parametrize any curve.

The examples in the book always use x=cos t and y=sint, but I know that there are more options.

That's presumably because the examples in the book are always about circles of radius 1! x²+ y²= cos² t+ sin² t= 1.

The example Office Shredder gave, x= acos t, y= b sin t, is an ellipse because
[tex]\frac{x^2}{a^2}+ \frac{y^2}{b^2}= \frac{a^2 cos^2 t}{a^2}+ \frac{b^2 sin^2 t}{b^2}= cos^2 t+ sin^2 t= 1[/itex]

There is no single way to determine parametric equations (as I said before, there are an infinite number of possibilities). Typically, one uses some kind of geometric property of the curve.