Finding an upper Darboux integral

I'm an undergraduate student, studing a first course in analysis. I
have a problem finding the upper Darboux integral of the following
function

f(x) = x ; for ratinal x
f(x) = 0 ; for irrational x

The problem asks me to find upper and lower Darboux integrals for f on
the interval [0,b]

This is what I have done in the lower Darboux integral,

Def:
P = {t_0 < t_1 < t_2 < ... < t_k < t_k-1 < ... < t_n} is a
partition
m(f,S) = inf {f(x) | for each element x of S}
M(f,S) = sup {f(x) | for each element x of S}
L(f,P) = Sum(from k=1 to n) m(f,[t_k-1,t_k]) * (t_k - t_k-1)
U(f,P) = Sum(from k=1 to n) M(f,[t_k-1,t_k]) * (t_k - t_k-1)
Lower Darboux integral
= L(f)
= sup { L(f,P) | P is a partition }
Upper Darboux integral
= U(f)
= inf { U(f,P) | P is a partition }

L(f,P) = Sum(from k=1 to n) m(f,[t_k-1 - t_k]) * (t_k - t_k-1)
= Sum(from k=1 to n) 0 * (t_k - t_k-1)
= 0 ; for every partition P
Thus I concluded that L(f) = 0


This is where I have the problem,

U(f,T) = Sum(from k=1 to n) M(f,[t_k-1,t_k]) * (t_k - t_k-1)
= Sum(from k=1 to n) t_k * (t_k - t_k-1)
= ... ; What's next?????

Then what is the upper Darboux integral?????

.