How to integrate a Weiner process that freezes at a determined time?

How to integrate a Weiner process that freezes at a determined time?

I would like to calculate the expected variance of the average of a Weiner
process from time 0 to time 1. The equation I believe I am trying to solve
is: $$ \int_{0}^{1} W_{t}^2 \ dt $$
My answer is:
$$ \int_{0}^{1} W_{t}^2 \ dt = \frac{t^3}{3} \biggr|_0^1 $$
This matches the numbers I get in a Monte Carlo simulation I coded for
what I am trying to do and the discrete time solution to this problem.
First, should there be any intermediate steps between the integral and the
solution? Did I state the problem correctly?
Now consider the case where the Weiner process "freezes" at time $s$ where
$0<s<1$. At time $s$ the Weiner process value becomes fixed, wherever it
is at that moment, and is essentially a horizontal line. More formally, I
am solving for:
$$ \int_{0}^{1} Q_{t}^2 \ dt $$
where
$$ Q_{t} = W_{t}: t<s \\ Q_{t} = W_{s}: t>s $$
How do I solve this problem?
I know from my discrete time solution and my Monte Carlo simulation that
the answer is:
$$ \int_{s}^{1} W_{t}^2 \ dt = \frac{t^3}{3} \biggr|_s^1 $$
or
$$ \int_{0}^{1} W_{t}^2 \ dt - \int_{0}^{s} W_{t}^2 \ dt = \frac{t^3}{3}
\biggr|_0^1 - \frac{t^3}{3} \biggr|_0^s $$
I don't quite understand why. What is the intuition behind this? How do I
properly explain this to others without any hand-waving?
This is for a work related issue and I don't have any coworkers I can talk
to for help. :( My stochastic calculus is a bit rusty. Any assistance
would be much appreciated.