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 Double Integrals Examples Example 2
 
del.icio.us Reddit Facebook StumbleUpon MySpace Twitter Search the VIAS Library  |  Index

Example 2

Find the double Riemann sum

12_multiple_integrals-24.gif

where D2 is the region

0 ≤ x ≤ 1, x2 ≤ y ≤ √x

and

Δx = ¼ Δy = 1/5.

12_multiple_integrals-23.gif

Figure 12.1.11

The circumscribed rectangle of D2 is the unit square. The partition and D2 are shown in Figure 12.1.11 and the partition points which actually belong to D2 are circled. The table shows the values of x2y at the partition points which belong to D2. It is a part of the table from Example 1.

x2y

y0 = 0

y1 = 1/5

y2 = 2/5

y3 = 3/5

y4 = 4/5

x0 = 0

0

x1 = ¼

1/80

2/80

x2 = ½

12/80

x3 = ¾

27/80

36/80

 

The double Riemann sum is

12_multiple_integrals-25.gif=

= 12_multiple_integrals-26.gif = 12_multiple_integrals-27.gif = 0.05375.

A similar computation with Δx = 1/10, Δy = 1/10 gives

12_multiple_integrals-28.gif
Home Multiple Integrals Double Integrals Examples Example 2

Last Update: 2010-11-25