Prelim1-3.Ex1-6%20in%2016.1

Prelim1-3.Ex1-6%20in%2016.1 - MULT IPLE 16 INTEGRATION 16.1...

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16 MULTIPLE INTEGRATION 16.1 Integration in Several Variables (ET Section 15.1) Preliminary Questions 1. In the Riemann sum S 8 , 4 for a double integral over R =[ 1 , 5 ]×[ 2 , 10 ] , what is the area of each subrectangle and how many subrectangles are there? SOLUTION Each subrectangle has sides of length 1 x = 5 1 8 = 1 2 ,1 y = 10 2 4 = 2 Therefore the area of each subrectangle is 1 A = 1 x 1 y = 1 2 · 2 = 1, and the number of subrectangles is 8 · 4 = 32. 2. Estimate the double integral of a continuous function f over the small rectangle R 0 . 9 , 1 . 1 1 . 9 , 2 . 1 ] if f ( 1 , 2 ) = 4. Since we are given the value of f at one point in R only, we can only use the approximation S 11 for the integral of f over R .For S 11 we have one rectangle with sides 1 x = 1 . 1 0 . 9 = 0 . 2 y = 2 . 1 1 . 9 = 0 . 2 Hence, the area of the rectangle is 1 A = 1 x 1 y = 0 . 2 · 0 . 2 = 0 . 04. We obtain the following approximation: ZZ R fdA S 1 , 1 = f ( 1 , 2 )1 A = 4 · 0 . 04 = 0 . 16 3. What is the integral of the constant function f ( x , y ) = 5 over the rectangle [− 2 , 3 2 , 4 ] ? The integral of f over the unit square R =[− 2 , 3 2 , 4 ] is the volume of the box of base R and height 5. That is, R 5 dA = 5 · Area ( R ) = 5 · 5 · 2 = 50 4. What is the interpretation of R f ( x , y ) if f ( x , y ) takes on both positive and negative values on R ? The double integral R f ( x , y ) is the signed volume between the graph z = f ( x , y ) for ( x , y ) R , and the xy -plane. The region below the -plane is treated as negative volume. 5. Which of (a) or (b) is equal to Z 2 1 Z 5 4 f ( x , y ) dydx ? (a) Z 2 1 Z 5 4 f ( x , y ) dx dy (b) Z 5 4 Z 2 1 f ( x , y ) The integral R 2 1 R 5 4 f ( x , y ) is written with dy preceding dx , therefore the integration is ±rst with respect to y over the interval 4 y 5, and then with respect to x over the interval 1 x 2. By Fubini’s Theorem, we may replace the order of integration over the corresponding intervals. Therefore the given integral is equal to (b) rather than to (a).
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This note was uploaded on 03/11/2008 for the course MATH 32B taught by Professor Rogawski during the Spring '08 term at UCLA.

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Prelim1-3.Ex1-6%20in%2016.1 - MULT IPLE 16 INTEGRATION 16.1...

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