There must be sense in your reasoning, because your numbers are coming out right (but why round down to 810? And strictly the bookies' odds for a probability of, e.g., 1/810 would be 809 to 1).
But if you can't explain your own reasoning, you may find it easier to ask: how many ways can the draw of six numbers include your brother's three?
And that's out of how many possibilities for the draw of six?
Get the probability and multiply it by £600 for the expected value of a ticket.
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!