differentation with respect to limits of integration

Question

I am interested in the time-dependent volume $V(t)$ given the width $w(x)$ and height $h(x,t)$---note that the width is constant in time but the height varies in time. Suppose the domain also varies in time $[0, x_g(t)]$. Therefore, the volume at time $t$ is

$V(t) = \int_0^{x_g(t)} w(x) h(x,t) dx$.

What is the derivative of the volume with respect to time? Mathematically, is there an expression for $\frac{dV}{dt}$?

Answer 1

Here's the answer so you can close it

$$ \frac{dV}{dt} = \frac{dx_g}{dt}w(x_g(t))h(x_g(t),t) + \int_0^{x_g(t)} w(x) \frac{\partial}{\partial t}h(x,t)\ dx $$