Electric field at the centre of a hemispherical shell

To find out the electric field at the centre of the hemispherical shell, I considered an elemental strip to be a ring, calculated the electric field due to it and integrated it as follows:

The expression of the field due to the ‘ring’ can be viewed here.
The answer I obtained matched the one my textbook stated. However, I am now beginning to doubt my solution. The elemental strips I have considered are not perfect rings. It won’t make a difference on an infinitesimal scale, but when we integrate the expression, I feel that the variations will get added up and cause a significant deviation from the correct answer, but that clearly is not the case. Is my solution wrong altogether? Please resolve this confusion.