Thread: Line Integral over a Plane Curve

Okay I'm having trouble parameterizing the curve C: x^2 + y^2 = 4 from (0,2) to (SQRT 2, SQRT 2) in the first quadrant in order to take the line integral for f (x,y) = x^2-y

I understand to put C into polar coordinates, and that r=2 but my key shows that r(t) = (2cost)i + (2sint)j 0<=t=<pi/2

and I have no idea how to get that r(t)...

Typically for a circle oriented counter clockwise you use

x= rcos(t) y = rsin(t) starting at (r,0)


Here though you are moving clockwise from(0,2) so switch the sine and cosine

use x = 2sin(t)

y= 2cos(t) and let t vary from 0 to pi/4

By the way

x= 2cos(t) y =2sin(t) 0<t<pi/2

is a counterclockwise parameterixation starting at (2,0) aand going to (0,2)

I see so any time you go counterclockwise as far as polar coordinates go, the x and y are flipped then?

you mean clockwise? It depends on where you start

Eg if you start at (-1,0) trvrling clockwise then x(t) =-cos(t) y(t) =sin(t)