Proving Integral is Bounded

Example

• Suppose $R$ is a rectangle
• Suppose $f(x,y)$ is Integrable and Bounded Function $|f(x,y)|\le M$ on $R$
• Note that this means $-MArea(R)\le \int {\int }_{R}f(x)dA\le MArea(R)$

Proving Bounded

Pick the Riemann Partition with $N$ sub-rectangles For upper bound: ${\sum }_{i,j=0}^{N-1}f(c_{ij})△x_i△y_i$

$<{\sum }_{i,j=0}^{N-1}fM△x_i△y_i$ $=M{\sum }_{i,j=0}^{N-1}f△x_i△y_i$ $=MArea(R)$ The lower bound is similar ${\sum }_{i,j=0}^{N-1}f(c_{ij})△x_i△y_i$ $>{\sum }_{i,j=0}^{N-1}f-M△x_i△y_i$ $=-M{\sum }_{i,j=0}^{N-1}f△x_i△y_i$ $=-MArea(R)$