# Finding the Line Integral

• May 27th 2011, 04:04 PM
Paymemoney
Finding the Line Integral
Hi
I am having trouble on working out the following question:

Given C is the anticlockwise circular path $\displaystyle x^2+y^2=4$, starting and ending at (0,2) evaluate the line integrals $\displaystyle \oint_C F.dr$ where $\displaystyle F = xyi+xyj$

This is what i have done

$\displaystyle r=xi + yj$
$\displaystyle dr = i + j$

$\displaystyle x=2cos(\theta)$, $\displaystyle y=2sin(\theta)$

$\displaystyle F = (4cos(\theta) sin(\theta))i + (4cos\theta sin\theta )j$

$\displaystyle \int_{0}^(2\pi) (4cos\theta sin\theta)i + (4cos\theta sin\theta )j \cdot i + j$

P.S
• May 27th 2011, 04:34 PM
Joanna
$\displaystyle dr\neq \hat{\imath} + \hat{\jmath}$
$\displaystyle dr = dx\hat{\imath} + dy \hat{\jmath}$

If
$\displaystyle x=2\cos\theta$, $\displaystyle y=2\sin\theta$

what are dx and dy?
• Jun 2nd 2011, 11:32 PM
Paymemoney
ok i have attempted this question again and this is what i have done but this is incorrect.

This is what i have have done:
$\displaystyle \int_0^{2\pi} 2cos(\theta)+2cos(\theta)2sin(\theta)*2cos(\theta)-2sin(\theta) d(\theta)$

$\displaystyle \int_0^{2\pi} 8cos(\theta)^2 * sin(\theta) - 8cos(\theta) * sin(\theta)^2 + 8cos(\theta)^2 * sin(\theta) - 8cos(\theta)* sin(\theta)^2$

$\displaystyle \int_0^{2\pi} 8(cos(\theta)^2 * sin(\theta) - cos(\theta) * sin(\theta)^2 + cos(\theta)^2 * sin(\theta) - cos(\theta)* sin(\theta)^2)$
• Jun 3rd 2011, 03:05 AM
HallsofIvy
Without parentheses, it is almost impossible to read what you have written. With $\displaystyle x= 2 cos(\theta)$, $\displaystyle y= 2 sin(\theta)$, $\displaystyle dx= -2 sin(\theta)dt$ and $\displaystyle dy= 2 cos(\theta)dt$. F(x,y) becomes $\displaystyle 4sin(\theta)cos(\theta)i+ 4 sin(\theta)cos(\theta)j$ and so the integral is
$\displaystyle 8\int_{\pi/2}^{5\pi/2} (-sin(\theta)cos^2(\theta)+ sin^2(\theta)cos(\theta))dt$
Personally, I would have used $\displaystyle x= 2sin(\theta)$, $\displaystyle y= 2cos(\theta)$ so I could integrate from 0 to $\displaystyle 2\pi$.