I am concerned that the following exercise has a typo. Can someone check to see if there is a typo in the conclusion of the following proposition?
Let $f$ and $g$ be two continuous functions such that $0 \leq m_1 \leq f(x) \leq M_1$ for any $x \in [a,b]$ and $0 \leq m_2 \leq g(y) \leq M_2$ for any $y \in [c,d]$. Show that the following inequality is true:
$$(m_1+m_2)(b-a)\color{blue}{(c-d)} \leq \int_a^b \int_c^d [f(x)+g(y)] \,dy\,dx \leq (M_1 + M_2)(b-a)\color{blue}{(c-d)}$$
I believe the typo is in blue in the above compound inequality, and it should be $(d-c)$, but I would like to confirm it.
Definitely it should be $d-c$. If you put $c-d$ there, then the inequality would be true only if you reverse the direction, so that it says greater-than-or-equal-to rather than less-than-or-equal-to.