Basics of Multiple Integrals and Applications  (continued)

 

 

The figure below shows the element of area,  dA, for a "single" integral case.

 

                          

 

 

 

 

 

Now consider  using a “double” integral for the above case. The element of area

dA is given by the small rectangle with dimensions  dy  by  dx.  i.e.  dA  = dy dx

                     

 

 

                                                       x2a     y2

So the total area becomes       A = ∫       ∫  dy  dx 

                                                       x1a    y1

 

 

Here the first integration is on the y-variable. You can picture this as “sweeping” the

element of area from y1 to y2  followed by “sweeping” the rectangle from x1a to x2a .

 

The same strategy (but more complicated) applies for calculating volumes between

 two intersecting surfaces.  In this case the element of volume is  dV  =  dx dy dz

if you "sweep" the volume first in x, then in y, and finally in z directions.

 

Click here for two examples involving double integrals.

 



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