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w9-C - Math 20C Multivariable Calculus Lecture 22 1 $...

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Math 20C Multivariable Calculus Lecture 22 1 Slide 1 & $ % Triple integrals On rectangular boxes. (Sec. 15.7) On simple domains, type I, II, and III. On arbitrary domains. Slide 2 & $ % Recall the Riemann sums and their limits Single variable functions: lim n →∞ n X i =0 f ( x * i x = Z x 1 x 0 f ( x ) dx. Two variable functions: lim n →∞ n X i =0 n X j =0 f ( x * i , y * j x Δ y = Z x 1 x 0 Z y 1 y 0 f ( x, y ) dxdy. Three variable functions: lim n →∞ n X i =0 n X j =0 n X k =0 f ( x * i , y * j , z * k x Δ y Δ z = Z x 1 x 0 Z y 1 y 0 Z z 1 z 0 f ( x, yz ) dxdydz.
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Math 20C Multivariable Calculus Lecture 22 2 Slide 3 & $ % Integrals in a rectangular box domain Theorem 1 Let f ( x, y, z ) be a continuous function on a rectangular boxed domain R = [ x 0 , x 1 ] × [ y 0 , y 1 ] × [ z 0 , z 1 ] . Then, Z Z Z R f dV = Z x 1 x 0 Z y 1 y 0 Z z 1 z 0 f ( x, y, z ) dzdydx. Furthermore, the integral does not change when performed in different order. Slide 4 & $ % Compute the integral of f ( x, y, z ) = xyz 2 on the domain R = [0 , 1] × [0 , 2] × [0 , 3] R = { ( x, y, z ) R 3 : 0 6 x 6 1 , 0 6 y 6 2 , 0 6 z 6 3 } . 1 2 y x 3 z
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Math 20C Multivariable Calculus Lecture 22 3 Slide 5 & $ % Notice the order of the integrations Z Z Z R f dV = Z 1 0 Z 2 0 Z 3 0 xyz 2 dzdydx, = Z 1 0 Z 2 0 xy 1 3 z 3 ˛ ˛ 3 0 dydx, = 27 3 Z 1 0 Z 2 0 xy dydx, = 9 Z 1 0 x 1 2 y 2 ˛ ˛ 2 0 dx, = 18 Z 1 0 x dx, = 9 . Slide 6 & $ % Triple integrals on simple regions Type I, means arbitrary shape only on the x variable. Type II means arbitrary shape only on the y variable. Type III means arbitrary shape only on the z variable.
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Math 20C Multivariable Calculus Lecture 22 4 Slide 7 & $ % For example, consider an integral type III Theorem 2 Let g 0 ( x, y ) 6 g 1 ( x, y ) be two continuous functions defined on a domain [ x 0 , x 1 ] × [ y 0 , y 1 ] . Let f ( x, y, z ) be a continuous function in D = 8 > > < > > : ( x, y, z ) R 3 : x 0 6 x 6 x 1 , y 0 6 y 6 y 1 , g 0 ( x, y ) 6 z 6 g 1 ( x, y ) 9 > > = > > ; .
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