While finding the Net Electric Field $\vec{E}$ at a point P (question 2.5 in https://i.stack.imgur.com/rbD4L.jpg ),I get that we are trying to simplify our problem by cancelling out the horizontal components here at the beginning ,but if we don't cancel the horizontal components at the beginning and go along with what I did for a second ,the components should cancel out eventually ,but I think I made some mistake here (please bear with me ,it's messy but I think you would understand what I've done here ) https://i.stack.imgur.com/n0Mb8.jpg ,where have I gone wrong ?(I have tried to find the electric field $\vec{E}$ with respect to a coordinate system ,that's why $r$ in the Coloumb's law is $\vec{r}=z\hat{z} - r\hat{r}$ ,is this wrong?
Edit: I think I've realised my mistake ,$\hat{r}$ is $cosθ \hat{i} + sinθ \hat{j}$ ,when I substitute this before integrating the $2^{nd}$ term of the expression ,I would've noticed the expression becoming zero as $sine$ and $cos$ are periodic functions and $0$ to $2π$ is their time period .Thank you for trying to help me .
