Theory

Introduction

Another perspective on integration extends the concept of the “unit of length” $dx$ to units of another measure. In calculus, one considers the “unit of area” such as the infinitesimal square $dxdy$ or the “unit of volume” such as the infinitesimal cube $dxdydz$.

From this point of view, the double integral ${\int }_c^d{\int }_a^{b}f(x,y)dxdy$ is the integral of the function $f(x,y)$ over the rectangular box $[a,b]\times [c,d]$, summing the infinitesimal quantities of area times height, $f(x,y)dxdy$. This sum can be written as the limit of a Riemann sum:

$${\iint }_{\mathcal{R}}f(x,y)dxdy=\underset{n,m\to \infty }{\lim }\sum _{i=1}^n\sum _{j=1}^{m}f(p_{ij})\Delta x_i\Delta y_j,$$

where $\mathcal{R}$ is the box $[a,b]\times [c,d]$, and $p_{ij}$ is a point that lives somewhere in the square $[x_i,x_{i+1}]\times [y_j,y_{j+1}]$ inside a grid over $\mathcal{R}$.

In line with this perspective on integration, sometimes the unit of area $dxdy$ is written suggestively as $dA$ with $A$ for “area”, and similarly $dxdydz$ can be written $dV$ with $V$ for “volume”.

Introduction (part 2)

Area of a region

By integrating the function $f(x,y)=1$ over a region, we obtain the area of the region:

$$A(\mathcal{R})={\iint }_{\mathcal{R}}dA={\iint }_{\mathcal{R}}dxdy.$$