I am trying to calculate E(\Int_0_T {W_s ds}), where W_s is a standard Brownian motion.

Now two approaches I can think of:

1) Take a partition of [0,T]. Calculate E(\Sum {W_t_i*(t_(i+1) - t_i)}) and take the limit as you shrink the size of the partition.

2) Calculate \Int_0_T { E(W_s)ds }.

However, for approach 1), its not clear what function would dominate the absolute value of the terms inside the E() for all possible partitions, and that it would have a finite expectation. So, interchanging limit and expectation is dicey.

For approach 2), Fubini's theorem would require me to know a-priori that E(\Int_0_T {|W_s|ds}) is finite, and I don't see how I could show that.

How can any of these approaches be fixed, if at all ? Or is there another way to solve the problem?