Explain how to relate the integral curvilinear of field vector with the area the region limited by a regular curve, simple and closed ? Apply the result and calculate the area of the ellipse:

Printable View

- Apr 27th 2009, 07:07 AMApprentice123Integral field vectorExplain how to relate the integral curvilinear of field vector with the area the region limited by a regular curve, simple and closed ? Apply the result and calculate the area of the ellipse:

- Apr 27th 2009, 10:05 AMCalculus26
Consider Green's Theorem

For F = (x-y) i +(x +y) j

d(x+y)/dx = 1 d(x-y)/dy = -1

The line integral then is 2times the double integral over R , or twice the area of the region enclosed by your ellipse

It is easier to compute this with the line integral than with a double integral.

so the area enclosed is 1/2 the line integral

parameterize C with x = acos(t) y = bsin(t)

and now you are good to go.

Remember you should get pi*ab - Apr 27th 2009, 10:59 AMApprentice123
- Apr 27th 2009, 11:06 AMCalculus26
No-- remember the area is 1/2 the line integral--so 0 t0 2pi is correct

if you use 0 to pi you don't have a closed curve - Apr 27th 2009, 11:10 AMApprentice123
- Apr 27th 2009, 11:13 AMCalculus26
No the answer is pi *ab The line integral is 2pi*ab and the AREA is 1/2 this

- Apr 27th 2009, 11:22 AMApprentice123
- Apr 27th 2009, 11:25 AMCalculus26
See my first post on this -- I explained it there

- Apr 27th 2009, 11:28 AMApprentice123
- Apr 27th 2009, 11:31 AMCalculus26
That would make sense wouldn't it ?

If F = f i +g j

And if dg/dx- df/dy is a constant k then Green's theorem gives k *Area - Apr 27th 2009, 11:38 AMApprentice123