Ex12-22%20%20in%2016.1

Ex12-22%20%20in%2016.1 - S E C T I O N 16.1 Integration in...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
SECTION 16.1 Integration in Several Variables (ET Section 15.1) 843 4 0.4 0.4 0.6 0.8 0.6 0.6 3 0.8 1.6 1.8 2.5 2.1 0.8 2 0.5 1.5 3.2 3.5 2.1 0.6 1 0.4 0.8 1.3 1.5 1.4 0.6 0 0.3 0.3 0.5 0.8 0.5 0.4 y x 012345 SOLUTION The function f ( x , y ) on the rectangle R = [0 , 5] × [0 , 4] gives the height of the mound at each point ( x , y ) R . We are given the values of f at some points. The volume is the double integral ZZ R f ( x , y ) dA ,whichwe estimate by an average of four Riemann sums. The computations are fairly tedious and we will not show them here. The Riemann sum using the lower-left corner of each rectange is 27 . 4, for lower-right it is 27 . 8, for upper-right we get 28 . 3, and for upper-left we get 27 . 8 The average is 27 . 825. 12. Figure 18 shows a grid with values of f ( x , y ) at sample points in each square. Estimate the double integral of f ( x , y ) over the rectangles: (a) [ 0 . 25 , 1 ]×[ 0 . 5 , 1 ] (b) [− 0 . 5 , 0 . 5 0 , 1 ] (c) [− 1 , 0 . 5 ]×[− 1 , 0 . 5 ] y x 1 1 1 1 5.5 5.2 5.4 4.5 4.1 4.5 5 6 4 4.9 4 3.5 3.5 3.9 4.5 5 3.3 3.5 3.3 2.3 2.9 3.5 4.2 5 2 2.5 3 2 2.5 2.7 3.5 4.5 1.3 2 2.3 2.4 3 3.5 4.2 4.4 0 1 1 2.5 3 3.4 4 4.5 100 . 500234 3 1.5 0.9 1 2012 FIGURE 18 Each subrectangle is a square of side 0 . 25, hence its area is: 1 A = 0 . 25 2 = 0 . 0625 We use the given data to estimate the double integral over the following squares. (a) R = [0 . 25 , 1] × [0 . 5 , 1]: The Riemann sum S 32 is the following sum: S 32 = ( 3 . 9 + 4 . 5 + 5 + 4 . 5 + 5 + 6 ) · 0 . 0625 1 . 81 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/11/2008 for the course MATH 32B taught by Professor Rogawski during the Spring '08 term at UCLA.

Page1 / 3

Ex12-22%20%20in%2016.1 - S E C T I O N 16.1 Integration in...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online