Vector Calculus | Part IA, 2006

Evaluate the line integral

α(x2+xy)dx+β(x2+y2)dy\int \alpha\left(x^{2}+x y\right) d x+\beta\left(x^{2}+y^{2}\right) d y

with α\alpha and β\beta constants, along each of the following paths between the points A=(1,0)A=(1,0) and B=(0,1)B=(0,1) :

(i) the straight line between AA and BB;

(ii) the xx-axis from AA to the origin (0,0)(0,0) followed by the yy-axis to BB;

(iii) anti-clockwise from AA to BB around the circular path centred at the origin (0,0)(0,0).

You should obtain the same answer for the three paths when α=2β\alpha=2 \beta. Show that when α=2β\alpha=2 \beta, the integral takes the same value along any path between AA and BB.

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