Chapter 5.1, Problem 44E

Let f and g be two continuous functions such that

in $0\le m_1\le f\left(x\right)\le M_1$for any $x\in \left[a,b\right]$ and

$0\le m_2\le g\left(y\right)\le M_2$, for any $y\in \left[c,d\right]$. Show that the

following inequality is true:

$\left(m_1+m_2\right)\left(b-a\right)\left(c-d\right)\le {\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{b}{\int }}\left[f\left(x\right)+g\left(y\right)\right]dydx\le \left(M_1+M_2\right)}\left(b-a\right)\left(c-d\right)}$.