Question

Use Green's Theorem to evaluate the line integral of F=⟨x^2,5x⟩
around the boundary of the parallelogram in the following figure (note the orientation).


With x0=5 and y0=5.

∫Cx^2dx+5xdy=

Answer 1

@phi or @ganeshie8

Answer 2

\[\oint_C x^2dx+5xdy = \iint_R curl(F) dA\]

Answer 3

find the curl of given vector field and setup the double integral ?

Answer 4

i'm not sure. i don't really understand. i got dp/dy=x^2y and dq/dz=5, but not sure what to do after that.

Answer 5

\(\large F = \langle x^2, 5x\rangle \)

\(curl = \dfrac{\partial }{\partial x }(5x) - \dfrac{\partial }{\partial y }(x^2) = 5-0 = 5 \)

Answer 6

he kinda sped through what green's theorem was. he's gonna finish today, but the assignment's also due today cause we have a test thursday.

Answer 7

oh, oops. i integrated that x^2 with y instead of differentiate. lol.

Answer 8

\[\large \begin{align} \oint_C x^2dx+5xdy &= \iint_R curl(F) dA \\~\\
&= \iint_R 5 dA \\~\\
&= 5 \iint_R dA \\~\\

\end{align}\]

Answer 9

you can work this geometrically because double integral of 1 simply gives you the area of the region

Answer 10

whats the area of given region ?

Answer 11

base times height. 5*5.

Answer 12

25

Answer 13

so is it just 5*(25)?

Answer 14

thats it

Answer 15

btw, if you have time, here's a good lecture on Green's theorem (for 2-D)
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-22-greens-theorem/

Answer 16

no.....webwork says it's wrong....

Answer 17

did you type in 125?

Answer 18

yes

Answer 19

try -125

Answer 20

oh, it's negative.

Answer 21

green's thm is sensitive to orientation of path, the path needs to be counterclockwise when you apply green's thm

Answer 22

ok. so if it's clockwise, it's negative. i'll put that in my notes. thanks

Answer 23

Yep! it will be there in hypothesis of green's thm in your textbook.. look it up :)