Using the definition of the complex integral, I was asked to evaluate the integral of the conjugate of a complex number $z$, denoted by $z*$ around the closed contour consisting of the portion of the parabola $y = x^2$ from $(0,0)$ to $(1,1)$ followed by the line segment from $z = 1 + i$ to $z = 0$.
My attempt at the problem was to first sketch the contour. Then I am essentially integrating $1 - i$ from $0$ to $1$, which is trivially $1-i$, but I'm not sure if this is correct. Any help would be greatly appreciated.
We wish to evaluate the contour integral
$$\displaystyle I=\oint_C z^* \,dz$$
where $C$ is the closed contour comprised of $(i)$ $C_1$, the parabolic path $y=x^2$ from $(0,0)$ to $(1,1)$ and $(ii)$ $C_2$, the line segment from $(1,1)$ to $(0,0)$.
One the parabolic path, we use the parameterization $x=t$ and $y=t^2$ so that $z^*=t-it^2$ and $dz=(1+i2t)\,dt$, to find
$$\int_{C_1}z^*\,dz=\int_0^1 (t-it^2)\,(1+i2t)\,dt\tag1$$
One the line segment from $(1,1)$ to $(0,0)$ we use the parameterization $x=t$ and $y=t$ to find
$$\int_{C_2}z^*\,dz=\int_1^0 (t-it)\,(1+i)\,dt \tag2$$
It is left as an exercise for the reader to evaluate the integrals on the right-hand sides of $(1)$ and $(2)$.