Question

Why are titration problems done with vigorous swirling?

Answer 1

It's done to make mixing of the reagents as complete as possible.

Explanation:

Example:
If you tirate a base into an acid with phenolphtaleine as an indicator, the indicator will go from colourless to purple where the base is dominant, i.e. the spot where the base hits the acid, even though the acid may be dominant in the rest of the liquid.
By swirling you spread the base throughout the whole of the liquid quickly, so the purple colour will only stay if the total amount of base is dominant over the total amount of acid.
And of course you want to see the moment (amount of base) when this just happens.

When you get near the equivalence point (but not quite there), the colour will be purple, but vanishes when your swirl. Go slowly now, drop by drop, and swirl between drops.