Question

Green's Theorem

Answer 1

I'm busy typing...

Answer 2

Use G's Theorem to find \[\int\limits_{C}^{}{F}.dr\] where F is given by

Answer 3

\[F = <y\cos{x}-xy\sin{x},xy+x\cos{x}>\] where C is the triangle from (0,0) to (0,4) to (2,0) to (0,0)

Answer 4

huh? is this algebra 2? wat chapter is this?!?!?!

Answer 5

Calc 3 (or 4, depending)

Answer 6

So here is where I started...

Answer 7

But.. now what? haha

Answer 8

@amistre64 @phi

Answer 9

Have you seen
http://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx

Answer 10

So, this isn't algebra 2?

Answer 11

@phi yip, I have, but I didn't see anything on this particular application of Green's. I will have a look again

Answer 12

Nope, I don't see that

Answer 13

I got y= - 2x+4 by determining the equation of the line from (2,0) to (4,0)

Answer 14

take a look at this ,might help u

Answer 15

your problem is close to Paul's first example

in your problem dr= < dx , dy >
and the problem is
\[\oint (y\cos{x}-xy\sin{x}) dx + (xy+x\cos{x})dy\]

Answer 16

Oh okay, I had no idea that I could write the F and dr in that way...

Then it would be a question just like any other by means of G?

Answer 17

you want to change the closed contour integral into a double integral over area
the limits will be x=0 to 2 and y=0 to - 2x+4

Answer 18

you want to find

\[ \int \int dQ/dx - dP/dy \ \ dA \]

Answer 19

I'm quickly doing it by hand, brb

Answer 20

I did it but got a different answer as the one in the answer section...

Answer 21

I got 16/3

Answer 22

that is the correct answer.

Maybe I just made a calculation err? I'm checking now

Answer 23

the dQ/dx - dP/dy simplified down to just y

Answer 24

I get 5.97


At this step, did I already make an error, or is it correct thus far? -->
\[\int\limits_{0}^{2}\int\limits_{0}^{-2x+4}y+x-x\sin{x}\text{ dy dx}\]

Answer 25

P is \( y\cos{x}-xy\sin{x} \)
what did you get for dP/dy

Q is \( (xy+x\cos{x}) \)
what do you get for dQ/dx

Answer 26

I got my P as y cos(x) - xy +x cos(x) and Q as xy+x cos(x)

I see that is where I made a mistake - my P is different

Answer 27

P is the first component of the vector F
\[ F = <y\cos{x}-xy\sin{x},xy+x\cos{x}> \]

Answer 28

Ahhh I see, I made a STUPID mistake... I wrote my P down incorrectly!

I'm going to do the whole problem again, brb

Answer 29

I'm back!

I got the same answer!

I see now, this work is actually quite easy, I just need to pay attention when re-writing and differentiating. Thanks!

Answer 30

Wait, wait, I just noticed that the answer sheet has the answer as - 16/3

where does the - come from?

Answer 31

You get +16/3 if you go around the curve counter clockwise. If the problem told you to go around the curve clockwise, you get the negative of 16/3

Answer 32

Okay thanks, maybe I misread that it said we should go CW.

But let's say we must go CW, how would the calculations differ? Do we just do this:\[-\int\limits_{?}^{?}\int\limits_{?}^{?}? d?d?\]?

Answer 33

if you go clockwise, everything is the same, except you put a minus sign out front of the double integrals

Answer 34

okay thanks so much!