How to evaluate an integral along a path?

[Guest]

Evaluate \(\displaystyle \int{(x^2 \,+\, y^2)\,dz}\) along the path \(\displaystyle C:\, x\,=\,t^2,\, y\,=\,\frac{1}{t},\,1\,\leq\, t\,\leq\, 3.\)

Any help would be greatly appreciated

 

[pka]

\(\displaystyle \L
\begin{array}{l}
\int {(x^2 + y^2 )dz = }\int (x^2 + y^2 )(dx + idy) = \int {(x^2 + y^2 )dx} + i\int {(x^2 + y^2 )dy} \\
\\
\int\limits_1^3 {\left( {t^4 + \frac{1}{{t^2 }}} \right)\left( {2tdt} \right)} + i\int\limits_1^3 {\left( {t^4 + \frac{1}{{t^2 }}} \right)\left( {\frac{{ - 1}}{{t^2 }}dt} \right)} \\
\end{array}\)