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

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aarnesBest ResponseYou've already chosen the best response.0
\[\int\limits_{?}^{?} (xy+xy)dy + (xy+x+y)dx \] Circle formula: \[x ^{2} + y ^{2}  ax = 0\]
 2 years ago

henkjanBest ResponseYou've already chosen the best response.0
just integrate: integral of (xy+xy)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)
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
Are you sure? I think you need to use polar coordinates.
 2 years ago

henkjanBest ResponseYou've already chosen the best response.0
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.
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
Ok, I'll try it and see how it goes.
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
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.
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
Here's my try to solve it using polar coordinates.
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
I switched from ax as limit to a+cos(theta) during integration. And also I used 2pi instead of (pi/2) to (pi/2)
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
That's a*cos(theta)...Typo.
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
I should've mentioned it. I'm using Green's theorem.
 2 years ago

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

TuringTestBest ResponseYou've already chosen the best response.2
If you wanted to change the bounds of theta to 02pi then you would at least have to make r vary from 0a/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.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.2
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.
 2 years ago

imranmeah91Best ResponseYou've already chosen the best response.1
since it is closed , consider green theorem
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.2
that's what we used, look at the scan above
 2 years ago

imranmeah91Best ResponseYou've already chosen the best response.1
(xy+xy)dy + (xy+x+y)dx N dy + M dx Nx= y1 My= x+1 \[\int \int Nx My dx dy\]
 2 years ago

imranmeah91Best ResponseYou've already chosen the best response.1
\[\int \int (yx ) dx dy\]
 2 years ago

imranmeah91Best ResponseYou've already chosen the best response.1
just integrate over circle
 2 years ago

imranmeah91Best ResponseYou've already chosen the best response.1
\[\int_0^{2\pi} \int_0^{ \frac{a}{2} cos(\theta)} a/2(sin(\theta))(cos(\theta)) r dr d\theta\]
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.2
^^^ 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\[\oint_c(xy+xy)dy+(xy+x+y)dx=\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{a\cos\theta}r^2(\sin\theta\cos\theta)drd\theta\]and got my answer above\[\frac{a^3\pi}{8}\]is there something wrong with that?
 2 years ago

aarnesBest ResponseYou've already chosen the best response.0
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!
 2 years ago
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