Let $f(z)=(x^2 + y)+ i(xy)$, $a=0$ and $b=1+i$. And let
$$A=\lbrace t+it^2: 0 \leq t \leq 1 \rbrace.$$
Prove
$$ \int_{A} f(z)= \frac{4}{15}+ i \frac{5}{4}$$.
And show
$$\int_{A} f(z) dz \neq \int_{B} f(z)dz,$$
where $B$ is the line segment joining the points $a$ and $b$.
We can see the set $A$ as a parametrized curve $\alpha(t)=t(1+i)$ for $t\in [0,1]$ and $B$ can be parametrized as the curve $\beta(t)=t(1+it)$. But seeing other complex integrals I cannot figure out how to integrate $\int_{A} f(z)$ and $\int_{B} f(z)$. Any guidance throught this will be aprecciated, the more explicit the better. Thanks
The first integral is$$\int_0^1f(t+it^2)\times(1+2ti)\,\mathrm dt,$$whereas the second one is$$\int_0^1f(t+ti)\times(1+i)\,\mathrm dt.$$Can you take it from here?