Mumbai University > COMPS > Sem 4 > Applied Mathematics 4

Marks : 05

Year : DEC 2015

$y = x$ is a straight line passing from origin at $45^\circ$ angle ![enter image description here][1] Let OA be the line from $z = 0$ to $z = 1 + i$ On OA $y =x \hspace {1cm} dy = dx$ $$Dz = dx +idy \space \space \space \space dz = dx + idx$$

$$= (1 + i) dx$$

And x varies from 0 to 1

$$\therefore I=\int\limits_0^1(x^2-ix)(1+i)dx$$

$=(1+i)[\dfrac {x^3}3-\dfrac {ix^2}2]_0^1\\ =\dfrac {(1+i)(2-3i)}6\\ =\dfrac {5-i}6$