Multiple (Riemann) integral over a rectangle

Let $R=\prod_{j=1}^n [a_j,b_j]\subseteq \mathbb{R}^n$ be a rectangle and let $f:R\to\mathbb{R}$ be bounded.

A partition $P$ of $R$ is a finite collection of subrectangles whose union is $R$ and whose interiors are pairwise disjoint (typically produced by partitioning each coordinate interval). For each subrectangle $Q\subseteq R$, let $M_Q=\sup_{x\in Q} f(x), \qquad m_Q=\inf_{x\in Q} f(x), \qquad \operatorname{vol}(Q)=\prod_{j=1}^n (\text{side length in coordinate }j).$ Define the upper sum and lower sum $U(f,P)=\sum_{Q\in P} M_Q\,\operatorname{vol}(Q), \qquad L(f,P)=\sum_{Q\in P} m_Q\,\operatorname{vol}(Q).$

The function $f$ is Riemann integrable over $R$ if $\sup_P L(f,P)=\inf_P U(f,P),$ and in that case the common value is the multiple (Riemann) integral of $f$ over $R$, denoted $\int_R f(x)\,dx.$

Multiple Riemann integrals generalize the usual one-dimensional integral to higher dimensions and are the starting point for Fubini's theorem and change-of-variables .

Examples:

• If $f(x)=1$ on $R$, then $\int_R 1\,dx=\operatorname{vol}(R)=\prod_{j=1}^n (b_j-a_j)$.
• If $R=[0,1]\times[0,1]$ and $f(x,y)=x+y$, then $\int_R (x+y)\,d(x,y)=\frac{1}{2}+\frac{1}{2}=1$.
• The indicator function of a rectangle is Riemann integrable; more generally, indicators of sets with sufficiently “small” boundary are integrable.