Basics of Multiple Integrals and Applications Click here for Double Integrals

 

 

 

1.

 

 

In a Nut Shell:  The integral under a curve, y(x), gives the area underneath the curve.

The differential area, dA, can be represented by  dA = y dx,  by  dA = x dy, or by

dA = dy dx.  This latter case leads to the double integral ∫ ∫ dy dx.  In a similar manner

the volume between two surfaces leads to a triple integral ∫ ∫ ∫dy dx dz.  In all cases

you need to determine the limits of integration.

 

 

 

 

 

 

 

 

 

 

 

2.

 

Recall that the total area under the curve, y = f(x), in Calculus 2 was given by:

 

                                          x2                      x2

                                   A = ∫  f(x) dx    =    ∫ y dx ------------------- (1)

                                          x1                     x1

 

The element of area, dA was visualized as a rectangle of width dx and height y

under the curve y = f(x). The total area then was the “sum” of each rectangle.

The region of integration extended from x1 to x2 .

                           

                       

 

 

 

 

 

 

3.

 

 

Next consider the area between two curves y1 = f1 (x) and y2 = f2 (x) where

the curve for y2 lies above y1 . Using the approach in Calculus 2, the area between

the curves is: 

 

                        x2a

                A = ∫  (y2 - y1 ) dx ------------------------------------- (2)

                       x1a

 

where x1a and x2a are the x-coordinates of the points of intersection of the

two curves.

 

Click here to continue with this case.

 


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