Multiple Integrals

Summary and Review

In this chapter, we saw that an ''integral of an integral'' is known as an iterated integral, whereas a double integral is the limit of double sums over ever finer partitions. However, if a region R is a type I region, then

f( x,y) dA =


q( x)

p( x) 
f( x,y) dydz
where x = a, x = b, y = p( x) , and y = q( x) are the boundaries of R. Similarly, a double integral over a type II region can be reduced to a type II iterated integral, and in both cases, the double integral yields the volume of the solid between z = f( x,y) and the xy-plane over the region R when f( x,y) 0. Other applications of the double integral include the area of a region, the mass and center of mass of a laminate, and the computation of probabilities for joint density functions.

Besides type I and type II regions, double integrals can be reduced to iterated integrals over regions in other coordinate systems. In that case, the differential is multiplied by the absolute value of the Jacobian determinant. Among the most important of these coordinate systems are polar coordinates, and indeed, the calculation of double integrals in polar coordinates is important in many applications of statistics.

Finally, the double integral concept can be extended to three or more integrals. Triple integrals often occur in association with densities, where a density is the measure of the amount of a physical quantity per unit volume of a geometric solid. Triple integrals in applications also occur frequently in either cylindrical or spherical coordinates, particularly when those applications involve regular solids such as spheres and right circular cylinders.

There are other ideas and concepts introduced in this chapter, and you should re-read the individual sections in addition to this summary.  The review materials are based on both the ideas above and some of those in the chapter not mentioned here.  Review questions and solutions are in web page form on the left and in pdf form on the right.  For maximum benefit, you should attempt to answer the questions before you look at the solutions.   

Web Pages Portable Document Format (PDF) 
  Review Questions Review Questions
  Review Solutions Review Solutions
  Maple Chapter Questions Maple Chapter Questions

You will need Acrobat reader in order to open and print the pdf files.


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