How to evaluate $x^2+iy^2$ on the line from 1 to i?

Question

I want to evaluate $$\int_\gamma f$$, where $f(x+yi) = x^2 + iy^2$, and $\gamma$ is the line joining $1$ to $i$.

I want to know the hint. How to calculate $f(z)$?

(The answer is $-\frac{2}{3}$).

Answer 1

Hint:

Substitute $y=1-x$ and calculate $$\int_{x=1}^{x=0}f(x,y)dz$$where $$dz=dx+idy=(1-i)dx\\f(x,y)=x^2+i(1-x)^2$$