Question A.4.2: Evaluate the line integral of A(x, y) = 2xyx + x^2y along th...
Evaluate the line integral of A(x, y) = 2xy\hat{x} + x^{2}\hat{y} along the three paths shown in Fig. A.6.
Along path 1 we have
\int_{C_{1}} A(x,y)⋅dl = \int_{0}^{y_{0}} 0 dy + \int_{0}^{x_{0}} 2x y_{0} dx
= x_{0}^{2} y_{0} .
Along path 2 we have
y = \frac{y_{0}}{x_{0}} x
and
dy = \frac{y_{0}}{x_{0}} dx,
with dl = dx\hat{x} + (y_{0}/x_{0}) dx\hat{y}. Thus, the integration reduces to
\int_{C_{2}} A(x,y) ⋅ dl = \int_{0}^{x_{0}} 2xy dx + \int_{0}^{x_{0}} x^{2} \frac{y_{0}}{x_{0}} dx
= 2\frac{y_{0}}{x_{0}} \int_{0}^{x_{0}} x^{2} dx + \frac{y_{0}}{x_{0}} \int_{0}^{x_{0}} x^{2} dx
= x_{0}^{2} y_{0} .
Last, along path 3 we obtain
\int_{C_{3}} A(x,y) ⋅ dl = \int_{0}^{x_{0}} 2x 0 dx + \int_{0}^{y_{0}} x_{0}^{2} dy
= x_{0}^{2} y_{0} .