The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Home Multiple Integrals Iterated Integrals Examples Example 3
 
del.icio.us Reddit Facebook StumbleUpon MySpace Twitter Search the VIAS Library  |  Index

Example 3

Let D be the region bounded by the curve xy = 1 and the line y = f - x. Find inequalities which describe D, and write down an iterated integral equal to ∫∫D f(x, y) dA.

Step 1

Sketch the region D as in Figure 12.2.3.

12_multiple_integrals-88.gif

Figure 12.2.3

Step 2

The line and curve intersect where

x(5/2 - x) = 1, x2 - 5/2 x + 1 = 0, (x - ½)(x - 2) = 0.

x = ½, x = 2.

For ½ ≤ x ≤ 2, the curve y = 1/x is below the line y = 5/2 - x. Therefore D is the region

½ ≤ x ≤ 2, 1/x ≤ y ≤5 /2 - x.

Step 3

The inequalities for x give the limits of the outside integral, and those for y give the limits of the inside integral. Thus

12_multiple_integrals-89.gif

Home Multiple Integrals Iterated Integrals Examples Example 3

Last Update: 2006-11-15