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$\vec{F}(x,y)=\alpha(a x \hat{i} + x y \hat{j})$ $A=(a,0)$ and $B=(a,a)$ \[\vec{W}_{OA} = \int_O^A \vec{F} \, d\vec{l} = \int_{x_O}^{x_A} \vec{F}(x,0). \, dx\hat{i} = \int_0^a \alpha(a x \hat{i} + x 0 \hat{j}) . \, dx \hat{i} = \int_0^a \alpha a x \, dx = [\alpha a x^2 / 2]_0^a = \alpha a^3 /2\] \[\vec{W}_{AB} = \int_A^B \vec{F} \, d\vec{l} = \int_{y_A}^{y_B} \vec{F}(a,y). \, dy\hat{j} = \int_0^a \alpha(a a \hat{i} + a y \hat{j}) . \, dy \hat{j} = \int_0^a \alpha a y \, dy = [\alpha a x^2 / 2]_0^a = \alpha a^3 /2\] such that $\vec{W}_{AB} = \alpha a^3$
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Sat, 18 Nov 2017 17:19 GMT