Question

Solve line integral (xy+x-y)dy + (xy+x+y)dx where the curve is a circle x^2+y^2-ax=0

Answer 1

\[\int\limits_{?}^{?} (xy+x-y)dy + (xy+x+y)dx \]
Circle formula: \[x ^{2} + y ^{2} - ax = 0\]

Answer 2

just integrate:

integral of (xy+x-y)dy = 1/2*x*y^2 + xy - 1/2y^2
integral of (xy+x+y)dx = 1/2*x^2*y + 1/2x^2 + yx

combining:
1/2*(x*y^2 - y^2+x^2*y + x^2) + 2*yx = 1/2*(-y^2+x^2) +2*x*y+1/2*(x^2*y+y^2*x)

Answer 3

Are you sure? I think you need to use polar coordinates.

Answer 4

You can always applly polar coordinates later on as well.... It might make it a little more cumbersome, but it will give the same value. But you can try, indeed.

Answer 5

Ok, I'll try it and see how it goes.

Answer 6

Well, I did same as you (nothing new) but I'm confused as to what limits to use. If converting to polar coordinates at start before integration I have one limit for angle theta 0 - 2pi and other for r 0 - a*cos(theta). With the result you got I don't know where to us these limits.

Answer 7

Here's my try to solve it using polar coordinates.

Answer 8

I switched from ax as limit to a+cos(theta) during integration. And also I used 2pi instead of -(pi/2) to (pi/2)

Answer 9

That's a*cos(theta)...Typo.

Answer 10

I should've mentioned it. I'm using Green's theorem.

Answer 11

Ok, leave the angle bounds as -pi/2 to pi/2. r varies from 0 to acos(theta). I got\[\oint_c(xy+x-y)dy+(xy+x+y)dx=-\frac{a^3\pi}{8}\]seemed rather straightforward, not sure about what the problem is.
Is this not the answer?

Answer 12

If you wanted to change the bounds of theta to 0-2pi then you would at least have to make r vary from 0-a/2, but that still won't work as far as I can see. You would need to change more things in order to do that; seems extraneous.

Answer 13

btw you were doing that last integral on your paper strangely. you could have integrated cos^3x*sinx right away, then tackled cos^4x with a reduction formula.

Answer 14

since it is closed , consider green theorem

Answer 15

that's what we used, look at the scan above

Answer 16

(xy+x-y)dy + (xy+x+y)dx

N dy + M dx

Nx= y-1

My= x+1

\[\int \int Nx- My dx dy\]

Answer 17

Nx=y+1

Answer 18

\[\int \int (y-x ) dx dy\]

Answer 19

just integrate over circle

Answer 20

\[\int_0^{2\pi} \int_0^{ \frac{a}{2} cos(\theta)} a/2(sin(\theta))-(cos(\theta)) r dr d\theta\]

Answer 21

^^^ that could work too, but why do you have a/2 in the integrand? shouldn't it be r first?
plus the circle you are integrating on imran is centered at (0,0), so won't that change things?

I did\[\oint_c(xy+x-y)dy+(xy+x+y)dx=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{a\cos\theta}r^2(\sin\theta-\cos\theta)drd\theta\]and got my answer above\[-\frac{a^3\pi}{8}\]is there something wrong with that?

Answer 22

Sorry for not answering for so long but I was away from keyboard. I too tried to do the problem using the r limit from 0 to a*cos(theta) and I got the same answer as TuringTest so I guess it's correct. Medal given. Thank you all for your input!