8.1 Introduction

Multiple integrals are used to find various engineering quantities:

• Areas (cost, heat losses, ...):
  $$\begin{eqnarray*} & \mbox{2D Cartesian:}\quad {\rm d}A = {\rm d}x {\rm d}y ... ...mbox{2D polar:}\quad {\rm d}A = \rho {\rm d}\rho {\rm d}\theta \end{eqnarray*}$$
  
  

• Volumes (weight, ...):
  $$\begin{eqnarray*} & \mbox{3D Cartesian:}\quad {\rm d}V = {\rm d}x {\rm d}y {\r... ...ad {\rm d}V = r^2 \sin \phi {\rm d}r {\rm d}\phi {\rm d}\theta \end{eqnarray*}$$
  
  

• Centroids (center of gravity, center of pressure, ...)
  
  $$\bar x = \int x {\rm d}A \: \bigg/ \int {\rm d}A \qquad \bar x = \int x {\rm d}V \: \bigg/ \int {\rm d}V \qquad$$
  
  

• Moments of inertia (solid body dynamics, center of pressure, ...)
  
  $$I_x = \int y^2 {\rm d}A \qquad I_0 = \int x^2 + y^2 {\rm d}A \qquad$$
  
  
  
  
  $$I_x = \int y^2 + z^2 {\rm d}V \qquad I_{xy} = - \int x y {\rm d}V \qquad$$
  
  

• ...

If the material is not homogenous, you may have to put an additional density-like factor in those integrals.

Procedure:

• Draw the region to be integrated over.
• When integrating, say $\int\int\int f(a,b,c)\; {\rm d}a {\rm d}b {\rm d}c$, you have to decide whether you want to do $a$, $b$, or $c$ first. Usually, you do the coordinate with the easiest limits of integration first.
• If you decide to do, say, $b$ first, (i.e. you want to integrate
  
  $$\int_{b_1}^{b_2} f(a,b,c)\;{\rm d}b$$
  
  
  first), the limits of integration $b_1$ and $b_2$ must be identified from the graph at arbitrary $a$ and $c$, and are normally functions of $a$ and $c$: $b_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $b_1(a,c)$, $b_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $b_2(a,c)$.
• After integrating over, say, $b$, the remaining double integral should no longer depend on $b$ in any way. Nor does the region of integration: redraw it without the $b$ coordinate. In other words, project it onto the $a,c$-plane. Then integrate over the next easiest coordinate in the same way.
• If you change integration variables from $a,b,c$ to $p,q,r$, the integral becomes $\int\int\int f(p,q,r) J\; {\rm d}p {\rm d}q {\rm d}r$ with the Jacobian
  
  $$J = \Big\vert \left\vert \begin{array}{ccc} \frac{\p... ...rtial c}{\partial r} \end{array} \right\vert \Big\vert$$
  
  
  Here the inner bars indicate the determinant of the matrix of derivatives and the outer bars the absolute value of that. (Sometimes it is easier to take the inverse of the Jacobian of the inverse transformation.)