Several times I come across the following transition between the quantity for some bounded Euclidean subset $U$
$$\int_0^T \int_{U} \dfrac{\partial}{\partial t} f(x,t)dx dt$$
to
$$\sup_{0<t<T} \int_{U} f(x,t)dx dt$$
I do not see how to go from the first to the second. Which one is greater? Or are they equal?
