Integrating the complex function $f(z) = x^2 + iy^2$

Question

Suppose $f(z) = x^2 + iy^2$ (where $z = x + iy$ ) and consider $C:z(t) = t + it,0 \leq t \leq 1$

Evaluate $\int_{C} f(z)dz$

I have tried solving this, I know that $\int_{C} f(z)dz = \int_a^b f(z(t))z'(t)dt$ for $a \leq t \leq b$

From $z(t)$ I can easily get $z'(t)$ But I struggle to obtain $f(z(t))$ because $f(z)$ is not expressed in terms of z.

Please help

Answer 1

Hint.

Note that $f(x+iy)=x^2+iy^2$. What do you have for $ f(t+it) $?

Answer 2

When someone writes $f(z) = x^2 + iy^2$ is is usually understood that $z = x + yi$, i.e. $x$ is the real part of $z$ and $y$ is the imaginary part of $z$.

So if $z = t +it$, can you find the real and imaginary parts of $z$?