It's also the name of my computer.
Integrity, to me, means wholeness. It means being all of one piece. It means walking your talk, living your values, embodying what is truest in you. It's deeper and purer than honesty.
In calculus, taking an integral means finding the area under a curve. All of it, all the bits and pieces. Finding the whole from the sum of its parts.
A set of data has integrity if it's complete, correct, and an accurate snapshot of the state of the data at a particular time.
So. Living in integrity means being all of one piece, living in harmony with my deepest values, expressing what is truest in me. It means living from my center, acting from that of God within me, flowing with what is best and purest and truest in me. It also means living passionately from that center, trusting my gut and my guide and living fully into my faith.