$\int_A f(x,t) \frac{\partial} {\partial t} g(x,t) dx dt = \int_{|x|\leq r}f(x,t) g(x,t) dx$?

Question

Let $f, g\in C^2(\mathbb R^n \times (0, \infty)).$ Put $A= \{ (x,t)\in \mathbb R^n \times (0, \infty): |x|<r, a<t<b \}$ ($r>0$).

Question:

Can we expect $$\int_A f(x,t) \frac{\partial} {\partial t} g(x,t) dx dt = \int_{|x|\leq r}f(x,t) g(x,t) dx$$?