Definite Integral of Kronecker product of matrices

Question

How to prove: $$\int_k^{k+1}\int_k^{k+1} (A⊗B) dxdy = \int_k^{k+1}\int_k^{k+1} A dx⊗\int_k^{k+1}\int_k^{k+1} Bdy $$, where k is an integer and A and B are matrices consisting of variables x and y respectively.

Answer 1

Since $(A,B)\rightarrow A\bigotimes B$ is bilinear, one has $\int_k^{k+1}\int_k^{k+1} (A⊗B) dxdy =\int_k^{k+1} A (x)dx⊗\int_k^{k+1} B(y)dy $.

EDIT. More generally $\int_{(x,y)\in [a,b]\times [c,d]} ( A(x)⊗B(y)) dxdy =\int_a^b A (x)dx⊗\int_c^d B(y)dy $.