ch15 - MULT IPLE 16 INTEGRATION 16.1 Integration in Several...

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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). 6. For which of the following functions is the double integral over the rectangle in Figure 16 equal to zero? Explain your reasoning. (a) f ( x , y ) = x 2 y (b) f ( x , y ) = 2 (c) f ( x , y ) = sin x (d) f ( x , y ) = e x x y 11 1 FIGURE 16
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SECTION 16.1 Integration in Several Variables (ET Section 15.1) 451 SOLUTION The double integral is the signed volume of the region between the graph of f ( x , y ) and the xy -plane over R . In (c) and (d) the function satisFes f ( x , y ) =− f ( x , y ) , hence the region below the -plane, where 1 x 0 cancels with the region above the -plane, where 0 x 1. Therefore, the double integral is zero. In (a) and (b), the function f ( x , y ) is always positive on the rectangle, so the double integral is greater than zero. Exercises 1. Compute the Riemann sum S 4 , 3 to estimate the double integral of f ( x , y ) = over R =[ 1 , 3 ]×[ 1 , 2 . 5 ] .Usethe regular partition and upper-right vertices of the subrectangles as sample points. The rectangle R and the subrectangles are shown in the following Fgure: x y 1 1.5 2 2.5 3 1 0 1.5 2 2.5 P 13 P 23 P 33 P 43 P 12 P 22 P 32 P 42 P 11 P 21 P 31 P 41 The subrectangles have sides of length 1 x = 3 1 4 = 0 . 5, 1 y = 2 . 5 1 3 = 0 . 5 1 A = 0 . 5 · 0 . 5 = 0 . 25 The upper right vertices are the following points: P 11 = ( 1 . 5 , 1 . 5 ) P 12 = ( 1 . 5 , 2 )
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This note was uploaded on 07/15/2009 for the course MATH 210 taught by Professor Hubscher during the Spring '08 term at University of Illinois at Urbana–Champaign.

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ch15 - MULT IPLE 16 INTEGRATION 16.1 Integration in Several...

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