Introduction

Multiple integrals:

• Areas (cost, ...):

  

• Volumes (weight, ...):

  

• Centroids (center of gravity, center of pressure, ...)

  

• Moments of inertia (solid body dynamics, center of pressure, ...)

  

  

• ...

Notes:

• Draw the region to be integrated over.
• When integrating, say , 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, ( first), the limits of integration b₁ and b₂ must be identified from the graph at arbitrary a and c, and are normally functions of a and c: b₁=b₁(a,c), b₂=b₂(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. Then integrate over the next easiest coordinate in the same way.