GATE 2017 EE – Question 38
Consider the line integral $I = \int_c (x^2 + iy^2)\,dz$, where $z = x + iy$. The line $c$ is shown in the figure below.
[Figure: The $x$-$y$ plane with a straight line from the origin $(0, 0)$ to the point $(1, i)$, that is $x = 1$ and $y = 1$.]
The value of $I$ is
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Correct answer: (B) $\frac{2}{3}i$
Explanation
On the line $y = x$ we have $z = x(1 + i)$, so $dz = (1 + i)\,dx$ and the integrand is $(x^2 + ix^2) = x^2(1 + i)$. Then $I = (1 + i)^2\int_0^1 x^2\,dx = \frac{2i}{3}$.