$V= \int_{t-\eta}^t \int_s^t f(v,s)\, dv\,ds$

Question

I wan t to know how to calculate $\frac{\partial V}{\partial t} $ of the following double integral.

$$V= \int_{t-\eta}^t \int_s^t f(v,s)\, dv\,ds$$

Answer 1

Hint. Use Leibniz integral rule, i.e. the one-dimensional differentiation under the integral sign (assuming that $f$ is sufficiently regular), $$\frac{dV}{dt}=\frac{d} {{dt}}\int\limits_{a\left( t \right)}^{b\left( t \right)} {F(t,s)} ds = F(t,b(t))b'(t)- F(t,a(t))a'(t) + \int_{a ( t)}^{b(t)} F_t(t,s)ds$$ where $F(t,s)=\int_s^t f(v,s) dv$ and therefore $F_t( t,s)=f(t,s)$.