[DEFINITIONS] Double integrals and Riemann sums

Double Riemann sum

For a partition of $R$ into rectangles $R_{ij}$ and tag points $(x_{ij}^*,y_{ij}^*)\in R_{ij}$ , the corresponding double Riemann sum is

\sum_{i=1}^{m}\sum_{j=1}^{n} f(x_{ij}^*,y_{ij}^*)\,\Delta A_{ij}.

When $f\ge 0$ , each term approximates the volume of a box above $R_{ij}$ . When $f$ takes both signs, the sum approximates a signed volume: contributions below the $xy$ -plane are subtracted.

Double integral over a rectangle

Let $f$ be bounded on $R=[a,b]\times[c,d]$ . If there is a number $I$ such that, for every $\varepsilon>0$ , some $\delta>0$ makes every tagged partition with $\lVert P\rVert<\delta$ satisfy

\left| \sum_{i=1}^{m}\sum_{j=1}^{n} f(x_{ij}^*,y_{ij}^*)\,\Delta A_{ij}-I \right|<\varepsilon,

then $f$ is Riemann integrable over $R$ and

\iint_R f(x,y)\,dA = \lim \sum_{i=1}^{m}\sum_{j=1}^{n} f(x_{ij}^*,y_{ij}^*)\,\Delta A_{ij}.

The notation

dA=dx\,dy=dy\,dx

records that area is being accumulated. It does not say which variable is integrated first.

Upper and lower double Riemann sums

For a partition $P$ of $R$ into rectangles $R_{ij}$ , the lower and upper double Riemann sums are

L(f,P)=\sum_{i,j}m_{ij}\,\Delta A_{ij}, \qquad U(f,P)=\sum_{i,j}M_{ij}\,\Delta A_{ij},

where $m_{ij}$ and $M_{ij}$ are the infimum and supremum of $f$ on $R_{ij}$ .

For a nonnegative $f$ , $L(f,P)$ is the volume of the tallest boxes on the partition that fit below the graph and $U(f,P)$ is the volume of the shortest boxes that reach above it. For a signed $f$ , the box heights are signed. Whenever the integral exists,

L(f,P)\le \iint_R f\,dA\le U(f,P).

Definitions from Double integrals and Riemann sums

Double Riemann sum

Double integral over a rectangle

Upper and lower double Riemann sums

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