Half of W Squared Is the Right Answer to the Wrong Sum
The integral of x dx is x squared over two, so the integral of W dW ought to be W(T) squared over two, and the only false step in that chain is the conclusion. A dissected square turns the Riemann sum into an identity exact at every partition, and the term that refuses to vanish is the total of the squared steps, which equals T rather than zero. The reflex answer is the exact value of the midpoint sum over the same partition, which is why it feels so solid.