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Use Green's Theorem to find the work done by the force $ \textbf{F}(x, y) = x(x + y) \, \textbf{i} + xy^2 \, \textbf{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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Johns Hopkins University

Missouri State University

Campbell University

Oregon State University

So we have to use green serum in order. Thio, solve this So we already know I and R J components of the vector. So all we have to do now is take the or the first thing we have to do is take the partial derivative of those components. So first we'll take the partial derivative of que with respect to X. So what that's ultimately going to be is, um why squared citizens, That's why squared and then the partial derivative of P with respect to why is just going to be X. So what we end up getting as a result of that is the double integral of white squared minus X. Yeah. Then, looking at the path followed by the particle, we see that the orientation as counterclockwise. So as a result, we can determine our bounds. Based on what they give us, we see that the bounds for X R 0 to 1 on the bounds for why are from 0 to 1 mine attacks. So because of that, we can make this do y the X, Then we'll focus on this first inter role. Yeah, okay. On this first inter rule, we'll end up giving us a one minus X cube over three, minus X plus X square and my fist. We take the integral again and we end up getting that. This is equal to a negative 1/12, as we see right there. So that tells us that the amount of work done was negative. 1/12. Andi, it depends on what the units are s so we can just say units.

California Baptist University

Vector Calculus