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Use Green's Theorem to find the work done by the force $\mathbf{F}(x, y)=x(x+y) \mathbf{i}+x y^{2} \mathbf{j}$ in moving a particle from the origin along the $x$ -axis to $(1,0),$ then along the line

segment to $(0,1),$ and then back to the origin along the $y$ -axis.

$-\frac{1}{12}$

Vector Calculus

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Campbell University

Harvey Mudd College

University of Nottingham

Boston College

this video, we're gonna determine, uh, using creates fear. Sorry. We're gonna determine the work done on a particle. Uh, well, we get along this curve. Just a try. So, first of all, the field that we heard is extensive. Possible life I had was exploits for jihad. So opinions except experts Why? But our Cuba's extents White square. So again, the work, it's just the light into growth. I thought the or which again, Reno. Greeks. There was just the double into role over the air over the domain. Dia que by the X minus one. And so now the derivative of Q respective X is just what squared and derivative p with respect. Why so p is X squared plus X y derivative with respect. So now let's see our why values They range from zero all the way up to that line. Okay. What What is the form developments, then your equation, That life Well, you know the wine work and to find the smoke the slope is just want to buy 11 divided by x two minus x one. So why desert zero minus one divided by what? So lying? Simply negative explosive. So this is our Why on our X values range for zero. Okay, so now we set up our double integral. So you set up the world to grow set up a double integral right here and now we're gonna take the derivative with respect to lies. Or it's just why Killed, divided by three minus seven X is closed it with respect, Ally. So the integral of X is just x Y amount are limits of integration are from zero to buy sex. So we put those values in right here. So we get one minus excuse Divided by three minus x times want might accept Just fix but square. So we take the intergroup on 01 of one bindis execute divide by X plus X squared the x So for first part starting to grow this that this over use it What our issue What's being with distant your right over So it just that you equal to one minus X So the inter grows with the use of the X So now we can write that chart weaken right that as negative being a little device by three you she's just going to negative u to the power for about 12. Just negative. One minus X to the power of 4 to 12. Okay, show we get tricky. Part of the integral Rex is just X squared Would like to into of expertise, just executed by three on our limits of integration on from Syria. So don't we just plug in one began zero minus what happened? Must prefer never forget zero. We get negative, but half my zero was so we just how negative one help plus 1/3 once 1 12 and that if we calling denominators, we get negative sips over 12 for 12 plus over 12. It's just negative, all right. And our orientation is count our voice So about lining to grow for the work. Ah, don't a particle. Well, moving it along that triangle just equal to negative for 12