Evaluate the line integral of f(x, y) = x^2 + y^2 along the three paths shown in Fig. A.6.

Question

Evaluate the line integral of f(x, y) = [latex]x^{2} + y^{2}[/latex] along the three paths shown in Fig. A.6.

Accepted Answer

Along path 1 we have

[latex]\int_{c_{1}}(x^{2} + y^{2})dl = (\int_{0}^{y_{0}} y^{2} dy)\hat{y} + (\int_{0}^{x_{0}} (x^{2} + y_{0}^{3} )dx)\hat{x}[/latex]

= [latex]x_{0} (\frac{x_{0}^{2} }{3}+y_{0}^{2} )\hat{x} +\frac{y_{0}^{2} }{3} \hat{y}.[/latex]

To integrate along path 2 we need to relate y and x. Along this path

y = [latex] \frac{y_{0}}{x_{0}}[/latex] x

and

dy = [latex] \frac{y_{0}}{x_{0}}[/latex] dx.

Therefore dl = dx [latex]\hat{x} + (y_{0} /x_{0})[/latex] dx[latex]\hat{y}[/latex]. The integration reduces to

[latex]\int_{c_{2}}(x^{2} + y^{2})dl = (\int_{0}^{x_{0}} (x^{2} + (\frac{y_{0}}{x_{0}} x)^{2}) (dx\hat{x} +\frac{y_{0}}{x_{0}} dx\hat{y})[/latex]

= [latex](1 + (\frac{y_{0}}{x_{0}})^{2})\frac{x_{0}^{3}}{3}\hat{x} + (1 + (\frac{y_{0}}{x_{0}})^{2}) \frac{y_{0} x_{0}^{2}}{3} \hat{y}.[/latex]

Last, along path 3 we have

[latex]\int_{c_{2}}(x^{2} + y^{2})dl = (\int_{0}^{x_{0}} x^{2} dx)\hat{x} + (\int_{0}^{y_{0}} (x_{0}^{2} +y^{2}) dy)\hat{y}[/latex]

= [latex]\frac{x_{0}^{3} }{3}\hat{x} + y_{0} (x_{0}^{2} +\frac{y_{0}^{2} }{3} ) \hat{y}[/latex].

Question A.4.1: Evaluate the line integral of f(x, y) = x^2 + y^2 along the ...