Double Integrals. MATH 375 Numerical Analysis. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Double Integrals
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1 Double Integrls MATH 375 Numericl Anlysis J. Robert Buchnn Deprtment of Mthemtics Fll 2013 J. Robert Buchnn Double Integrls
2 Objectives Now tht we hve discussed severl methods for pproximting definite integrls of the form f (x) dx we turn our ttention to double integrls of the form f (x, y) da. R In this lesson we will discuss qudrture methods which cn be pplied to the cse when R = {(x, y) x b, c y d}, nd R = {(x, y) x b, c(x) y d(x)}. J. Robert Buchnn Double Integrls
3 Objectives Now tht we hve discussed severl methods for pproximting definite integrls of the form f (x) dx we turn our ttention to double integrls of the form f (x, y) da. R In this lesson we will discuss qudrture methods which cn be pplied to the cse when R = {(x, y) x b, c y d}, nd R = {(x, y) x b, c(x) y d(x)}. Generliztions to polr coordintes nd to triple integrls cn lso be mde. J. Robert Buchnn Double Integrls
4 Integrting Over Rectngulr Region (1 of 2) Suppose R = {(x, y) x b, c y d}, then d [ b ] d f (x, y) da = f (x, y) dy dx = f (x, y) dy dx. R c c J. Robert Buchnn Double Integrls
5 Integrting Over Rectngulr Region (1 of 2) Suppose R = {(x, y) x b, c y d}, then d [ b ] d f (x, y) da = f (x, y) dy dx = f (x, y) dy dx. R c c Let m be n even integer nd pply the Composite Simpson s rule to the inner integrl. J. Robert Buchnn Double Integrls
6 Integrting Over Rectngulr Region (2 of 2) d c f (x, y) dy = k f (x, y 0 ) m/2 1 (d c) 180 k 4 4 f (x, µ) y 4 m/2 f (x, y 2j ) + 4 f (x, y 2j 1 ) + f (x, y m ) where k = d c m y j = c + j k for j = 0, 1,..., m µ [c, d] J. Robert Buchnn Double Integrls
7 Outer Integrtion (1 of 2) Now integrte with respect to x. d = f (x, y) dy dx k 3 c f (x, y 0 ) + 2 m/2 1 (d c) 180 k 4 4 f (x, µ) dx y 4 m/2 f (x, y 2j ) + 4 f (x, y 2j 1 ) + f (x, y m ) d J. Robert Buchnn Double Integrls
8 Outer Integrtion (2 of 2) d c = k 3 f (x, y) dy dx + 4 f (x, y 0 ) dx + 2 m/2 (d c) 180 k 4 m/2 1 f (x, y 2j 1 ) dx + 4 f (x, µ) dx y 4 f (x, y 2j ) dx f (x, y m ) dx J. Robert Buchnn Double Integrls
9 Outer Integrtion (2 of 2) d c = k 3 f (x, y) dy dx + 4 f (x, y 0 ) dx + 2 m/2 (d c) 180 k 4 m/2 1 f (x, y 2j 1 ) dx + 4 f (x, µ) dx y 4 f (x, y 2j ) dx f (x, y m ) dx Fix vlue of j nd pply the Composite Simpson s rule to the integrl f (x, y j ) dx. J. Robert Buchnn Double Integrls
10 Integrtion with Fixed j If n is n even integer nd then f (x, y j ) dx h = b n x i = + i h for i = 0, 1,..., n ξ j [, b] = h f (x 0, y j ) n/2 1 i=1 (b ) 180 h4 4 f x 4 (ξ j, y j ) n/2 f (x 2i, y j ) + 4 f (x 2i 1, y j ) + f (x n, y j ) i=1 J. Robert Buchnn Double Integrls
11 Qudrture Formul d f (x, y) dy dx c hk n/2 1 n/2 f (x 0, y 0 ) + 2 f (x 2i, y 0 ) + 4 f (x 2i 1, y 0 ) + f (x n, y 0 ) 9 i=1 i=1 m/2 1 m/2 1 n/2 1 m/2 1 n/2 + 2 f (x 0, y 2j ) + 2 f (x 2i, y 2j ) + 4 f (x 2i m/2 m/2 + 4 f (x 0, y 2j 1 ) f (x 0, y m ) + 2 n/2 1 i=1 i=1 n/2 1 i=1 m/2 n/2 i=1 f (x 2i, y 2j 1 ) + 4 f (x 2i 1 i=1 n/2 f (x 2i, y m ) + 4 f (x 2i 1, y m ) + f (x n, y m ) i=1 J. Robert Buchnn Double Integrls
12 Error The error term for the double integrl Composite Simpson s rule qudrture formul tkes the form of [ ] (d c)(b ) E(f ) = h 4 4 f 180 x 4 (ξ, µ) + k 4 4 f y 4 (ˆξ, ˆµ) J. Robert Buchnn Double Integrls
13 Exmple (1 of 2) Let m = 4 nd n = 6 nd use the Composite Simpson s rule to pproximte the double integrl: π/4 0 sin x+cos 0 π/3(2y 2 x) dy dx = π [ 6 + π(4 ] 2 5) z Π 24 Π 12 Π x 8 Π 6 5 Π 24 Π Π 3 4 J. Robert Buchnn Double Integrls Π 4 0 Π 12 Π 6 y
14 Exmple (2 of 2) π/4 0 0 π/3 (2y sin x + cos 2 x) dy dx Absolute error: π [ 6 + π(4 ] 2 5) Error bound: E(f ) mx c)(b ) (d (ξ,µ),(ˆξ,ˆµ) R 180 = (π/3)(π/4) ( π ) 4 mx = (ξ,µ),(ˆξ,ˆµ) R [ h 4 4 f π J. Robert Buchnn Double Integrls x 4 (ξ, µ) + k 4 4 f ˆµ)] y 4 (ˆξ, 8(cos 2 x sin 2 x) + 2y sin x
15 Double Integrls with Gussin Qudrture We my lso use Gussin qudrture formuls to pproximte double integrls. Consider the double integrl: d = = f (x, y) dy dx c f ) (d c)u + c + d d c du dx 2 2 ( ) (b )v + + b (d c)u + c + d, 2 2 ] du dv ( x, [ f (b )(d c) 4 J. Robert Buchnn Double Integrls
16 Gussin Qudrture Formul For the ske of convenience we will use the sme precision, n, for ech stge of the integrtion. d c f (x, y) dy dx (b )(d c) 4 n n ( (b )rn,i + + b c n,i c n,j f 2 i=1, (d c)r ) n,j + c + d 2 J. Robert Buchnn Double Integrls
17 Exmple Using the double integrl form of Gussin qudrture with n = 4 we my pproximte π/4 0 0 π/3 (2y sin x + cos 2 x) dy dx Absolute error: π [ 6 + π(4 ] 2 5) J. Robert Buchnn Double Integrls
18 Integrting Over Non-rectngulr Regions Consider the double integrl f (x, y) da = R d(x) c(x) f (x, y) dy dx. Suppose we use the Composite Simpson s rule to pproximte the inner integrl. J. Robert Buchnn Double Integrls
19 Integrting Over Non-rectngulr Regions Consider the double integrl f (x, y) da = R d(x) c(x) f (x, y) dy dx. Suppose we use the Composite Simpson s rule to pproximte the inner integrl. d(x) c(x) f (x, y) dy k(x) 3 f (x, c(x)) + 2 m/2 1 f (x, c(x) + 2jk(x)) m/2 + 4 f (x, c(x) + (2j 1)k(x)) + f (x, d(x)) where m is n even integer nd k(x) = J. Robert Buchnn Double Integrls d(x) c(x). m
20 Simplifiction (1 of 2) To mke the nottion of the next step simpler we will mke the following chnge to the nottion. F(x) = k(x) m/2 1 f (x, c(x)) + 2 f (x, c(x) + 2jk(x)) 3 m/2 +4 f (x, c(x) + (2j 1)k(x)) + f (x, d(x)) J. Robert Buchnn Double Integrls
21 Simplifiction (2 of 2) This implies d(x) c(x) f (x, y) dy k(x) 3 f (x, c(x)) + 2 m/2 1 f (x, c(x) + 2jk(x)) m/2 + 4 f (x, c(x) + (2j 1)k(x)) + f (x, d(x)) = F(x) J. Robert Buchnn Double Integrls
22 Simplifiction (2 of 2) This implies d(x) c(x) f (x, y) dy k(x) 3 f (x, c(x)) + 2 m/2 1 f (x, c(x) + 2jk(x)) m/2 + 4 f (x, c(x) + (2j 1)k(x)) + f (x, d(x)) = F(x) Now we my pproximte the outer integrl. J. Robert Buchnn Double Integrls
23 Outer Integrl Now we choose n to be n even integer nd pply the Composite Simpson s rule with h = b n. d(x) = c(x) f (x, y) dy dx k(x) 3 m/2 +4 F(x) dx h F() f (x, c(x)) + 2 m/2 1 f (x, c(x) + 2jk(x)) f (x, c(x) + (2j 1)k(x)) + f (x, d(x)) dx n/2 1 i=1 n/2 F( + 2ih) + 4 F( + (2i 1)h) + F(b) i=1 J. Robert Buchnn Double Integrls
24 Exmple (1 of 2) Consider the double integrl, 1 2x 0 x (x 2 + y 3 ) dy dx = 1 1 Using the composite Simpson s rule with n = m find the smllest vlues of n nd m required to estimte the double integrl to within 10 4 of its ctul vlue. 2 Estimte the double integrl. J. Robert Buchnn Double Integrls
25 Exmple (2 of 2) Since we hve no formul for the error term in this cse we will use the composite Simpson s rule with n = m = 2, 4,... until successive pproximtions differ by less thn n = m Estimte J. Robert Buchnn Double Integrls
26 Gussin Qudrture for Generl Regions (1 of 2) Consider the double integrl: d(x) c(x) f (x, y) dy dx We cn pply Gussin qudrture to the inner integrl. J. Robert Buchnn Double Integrls
27 Gussin Qudrture for Generl Regions (1 of 2) Consider the double integrl: d(x) c(x) f (x, y) dy dx We cn pply Gussin qudrture to the inner integrl. d(x) c(x) = f (x, y) dy 1 1 f ( x, d(x) c(x) 2 ) (d(x) c(x))u + c(x) + d(x) d(x) c(x) du 2 2 n ( c n,j f x, (d(x) c(x))r ) n,j + c(x) + d(x) 2 J. Robert Buchnn Double Integrls
28 Gussin Qudrture for Generl Regions (2 of 2) Therefore d(x) c(x) f (x, y) dy dx d(x) c(x) 2 n c n,j f ( x, (d(x) c(x))r ) n,j + c(x) + d(x) 2 Now we pply Gussin qudrture to the remining integrl. J. Robert Buchnn Double Integrls
29 Algorithm INPUT, b; positive integers m, n. STEP 1 Set h 1 = (b )/2; h 2 = (b + )/2; J = 0. STEP 2 For i = 1, 2,..., m do STEPS 3 5. STEP 3 Set JX = 0; x = h 1 r m,j + h 2 ; d = d(x); c = c(x); k 1 = (d 1 c 1 )/2; k 2 = (d 1 + c 1 )/2. STEP 4 For j = 1, 2,..., n set y = k 1 r n,j + k 2 ; Q = f (x, y); JX = JX + c n,j Q. STEP 5 Set J = J + c m,i k 1 JX. STEP 6 Set J = h 1 J; OUTPUT J. J. Robert Buchnn Double Integrls
30 Exmple (1 of 2) Use Gussin qudrture with n = 4 to pproximte the double integrl e x 2 1 x ln(x y) dy dx. 3 z y 2.0 x J. Robert Buchnn Double Integrls
31 Exmple (2 of 2) e x 2 1 x ln(x y) dy dx Absolute error: e x 2 ln(x y) dy dx x J. Robert Buchnn Double Integrls
32 Homework Red Section 4.8. Exercises: 1b, 5b J. Robert Buchnn Double Integrls
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