How are limits of integration of a double integral affected when that integral is negated?
I understand that $\int_a^b(1)dx = -\int_b^a(1)dx$. How would this identity translate to a double integral, say $\int_a^b\int_c^d(1)dydx$?
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You flip the sign for every switch in limits.
$$\int_a^b \int_c^d dy\,dx = -\int_a^b \int_d^c dy\,dx = - \int_b^a \int_c^d dy\,dx = \int_b^a \int_d^c dy\,dx$$
When the region of integration is a square $[a..b]\times [c..d]$ then the double integral can be evaluated independently. I mean: $$\int_a^b\int_c^d dxdy=\int_a^b dx\int_c^ddy$$