# Homework Help: Gauss-Green Cardioid problem not coming out right

1. Jul 15, 2012

### delta59

1. The problem statement, all variables and given/known data
Here is the region R plotted in the xy plane by the functions
X(t)=cos(t)(1-cos(t))
Y(t)=sin(t)(1-cos(t))

go with
f(x,y)=3+y-x and g(x,y)=3-x

calculate ∫∫_R f(x,y) dxdy and ∫∫_R g(x,y) dxdy

2. Relevant equations

3. The attempt at a solution
I know I need to use the Gauss-Green formula so

X(t)=cos(t)(1-cos(t)
X'(t)=-sin(t)(-cos(t)+1)+cos(t)sin(t)

Y(t)=sin(t)(1-cos(t)
Y'(t)=cos(t)(-cos(t)+1)+sin(t)^2

a=0
b=2∏

f(x,y)=3+y-x

m(x,y)=0
n(x,y)=∫f(s,y)ds=-.5x^2+3x+xy

∫(m(X(t),Y(t))X'(t)+n(X(t),Y(t))Y'(t)) dt from a to b

Now I know this is a cardioid with a area of 3∏/2 but when I put everything in I get 18.06. Is my a and b wrong because I swear I have done everything else right

Last edited by a moderator: Jul 15, 2012
2. Jul 15, 2012

### vela

Staff Emeritus
The integral won't come out to be equal to the area of the cardioid. Why do you think it does? Your answer of 18.06 is correct.

3. Jul 15, 2012

### delta59

lol wow I feel stupid, its measuring a 3d space formed by the cardiod and the plane of f(x,y) and g(x,y). I get it now