Question

Lineintegrals and double integrals: f(x,y)=x^2y+yx for (x,y) element of R, where R is the region between y=0, x=4 and y=x. Evaluate integral with the limits of R f(x,y) dS. I'm suspecting I'll have to convert this into a double integral, but not sure what to do with the given function and not sure about the limits either

Answer 1

Do I apply Green on the equation?

Answer 2

show your attempt (sketch the region atleast)
its hard to get hang of problem just by reading it once :S

Answer 3

I haven't sketched the function, but I have a sketch of the region in pencil

Answer 4

So basically you want to evaluate the double integral of f(x,y) in the above region. Is that correct ?

Answer 5

\[\large \iint_R ( x^2y+yx) dxdy\]

Answer 6

I think I have to apply Green's theorem on the function to make it into a double integral. I can do that because the region is closed loop.

Answer 7

Green's theorem is used to compute \(\text{work}\) done along a closed loop.

What exactly are you trying to do here ? question did not give any vector field or it was not even asking for work :/

Answer 8

It would be helpful if you can take a snapshot of actual question and attach

Answer 9

\[\int\limits_{R}^{}f(x,y) dS=\int\limits_{R}^{}x^2y dx+yx dy\]

Green is for vector fields, not necessarily work. I don't think a snapshot will be of any help, unless you can read Dutch. No drawings. But the b) part of the question says: what would happen if you change the integration order around? Which can only mean getting a double integral.

The exercise sheets are about line integrals and double integrals. Surface/Area integrals is the next sheet. The region is closed and can be seen as a closed field.

Answer 10

you're not given any vector field here, so it makes no sense to talk about work or green's theorem here

Answer 11

The integral would become
\[\int\limits_{}^{} \int\limits_{D}^{}(2xy-x) dx dy\]

Answer 12

what you're given is a function of two variables f(x,y) and a region to integrate over, so all you need to is to integrate. thats all i think

Answer 13

a part question is evaluate
\[\int\limits_{R}^{}f(x,y) dS\]

Answer 14

strictly speaking, it should be asked as : \(\large \iint\limits_R f(x,y) dS\)

Answer 15

Well, that's the problem: that's not what the exercise sheet says.

Answer 16

region => doible integral
path => line integral
space => triple integral

Answer 17

Then they must have made a typo or something

Answer 18

possible, did you get why this is not a `work/flux/green's theroem` problem ?

Answer 19

Yes

Answer 20

why ?

Answer 21

It wasn't said it's a vector field

Answer 22

exactly !
vector field requires two components : F = <M, N>

but in your question there is no vector field, so you shouldn't even think about `work/green's theorem` here