Accumulate over two coordinates

Definition For real numbers $a < b$ and $c < d$, the closed rectangle $R = [a, b] \times [c, d]$ is the set of points $(x, y)$ with $a \le x \le b$ and $c \le y \le d$. Its sides are parallel to the axes and its area is $(b - a)(d - c)$. On a rectangular region every bound is a constant. The range of $y$ is the same interval $[c, d]$ for every $x$, and the range of $x$ is $[a, b]$ for every $y$.

Definition Let $f$ be a bounded function on $R = [a, b] \times [c, d]$. Points $a = x_0 < x_1 < \dots < x_m = b$ and $c = y_0 < y_1 < \dots < y_n = d$ cut $R$ into the cells $R_{ij} = [x_{i-1}, x_i] \times [y_{j-1}, y_j]$, with areas $\Delta A_{ij} = \Delta x_i \, \Delta y_j$. Pick a sample point $(x_{ij}^*, y_{ij}^*)$ in each cell and form the double Riemann sum

Suppose these sums approach one number $I$ as the longest cell diagonal shrinks to $0$, whatever sample points are chosen. Then $f$ is integrable on $R$, and $I$ is its double integral,

Every function that is continuous on $R$ is integrable there. The symbol $dA$ stands for the area of a small cell of the base.

Definition Let $f$ be integrable on $R$ with $f(x, y) \ge 0$ at every point of $R$. The solid of points $(x, y, z)$ with $(x, y)$ in $R$ and $0 \le z \le f(x, y)$ has volume

If $f$ takes negative values, the integral is a signed volume. The space between the graph and the base counts as positive where the graph lies above the $xy$ plane and as negative where it lies below. The volume between the graph and the base is then the integral of $\lvert f \rvert$.

Definition For a function $f$ on $R = [a, b] \times [c, d]$, the iterated integral in the order $dy\,dx$ is

The bracket is the inner integral, taken first in $y$ from $c$ to $d$. Its value is then integrated by the outer integral, in $x$ from $a$ to $b$. The differentials are read from the inside out, and each pair of limits belongs to the differential in the matching position, the innermost limits to the innermost differential. The iterated integral is defined when the inner integral exists for every $x$ in $[a, b]$ and its value is an integrable function of $x$. The order $dx\,dy$ is defined in the same way, with the roles of $x$ and $y$ exchanged. If $f$ is continuous on $R$, both iterated integrals exist and both equal $\iint_R f\,dA$. This is Fubini’s theorem for a rectangle.

Definition In the inner integral $\int_c^d f(x, y)\,dy$, the outer variable $x$ is held fixed. It is treated as a constant while $y$ runs from $c$ to $d$, so a term without $y$ integrates like a constant, and the evaluation substitutes $y = d$ and $y = c$ only. The result

is a function of $x$ alone. When $f \ge 0$, the number $A(x)$ is the area of the cross section of the solid by the plane perpendicular to the $x$ axis at that $x$. The outer integral $\int_a^b A(x)\,dx$ then adds these areas, as in the slicing method for volumes.