w8-C - Math 20C Multivariable Calculus Lecture 21 1 Slide 1...

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Unformatted text preview: Math 20C Multivariable Calculus Lecture 21 1 Slide 1 & $ % Double integrals on regions (Sec. 15.3) Regions function of y . Regions function of x . Properties of double integrals. Slide 2 & $ % Regions functions of y Theorem 1 (Type I) Let g ( x ) , g 1 ( x ) be two continuous functions defined on an interval [ x , x 1 ] , and such that g ( x ) g 1 ( x ) . Let f ( x, y ) be a continuous function in D = { ( x, y ) IR 2 : x x x 1 , g ( x ) y g 1 ( x ) } . Then, the integral of f ( x, y ) in D is given by Z Z D f ( x, y ) dxdy = Z x 1 x " Z g 1 ( x ) g ( x ) f ( x, y ) dy # dx. Math 20C Multivariable Calculus Lecture 21 2 Slide 3 & $ % Example: Type I Find the R R D f ( x, y ) dxdy for f ( x, y ) = x 2 + y 2 , D = { ( x, y ) IR 2 : 0 x 1 , x 2 y x } . Slide 4 & $ % Z Z D f ( x, y ) dxdy = Z 1 Z x x 2 ( x 2 + y 2 ) dy dx, = Z 1 x 2 ( y | x x 2 ) + 1 3 ( y 3 | x x 2 ) dx, = Z 1 x 2 ( x- x 2 ) + 1 3 ( x 3- x 6 ) dx, = Z 1 x 3- x 4 + 1 3 x 3- 1 3 x 6 dx, = 1 4 x 4 | 1- 1 5 x 5 | 1 + 1 12 x 4 | 1- 1 21 x 7 | 1 , = 1 3- 1 5- 1 21 = 9 3 5 7 . Math 20C Multivariable Calculus Lecture 21 3 Slide 5 & $ % Regions functions of x Theorem 2 (Type II) Let h ( y ) , h 1 ( y ) be two continuous functions defined on an interval [ y , y 1 ] , and such that h ( y ) h 1 ( y ) . Let f ( x, y ) be a continuous function in D = { ( x, y ) IR 2 : h ( y ) x h 1 ( y ) , y y y 1 } . Then, the integral of f ( x, y ) in D is given by Z Z D f ( x, y ) dxdy = Z y 1 y " Z h 1 ( y ) h ( y ) f ( x, y ) dx # dy. Slide 6 & $ % Example type II Find the R R D f ( x, y ) dxdy for f ( x, y ) = x 2 + y 2 , D = { ( x, y ) IR 2 : 0 x 1 , x 2 y x } . Math 20C Multivariable Calculus Lecture 22 4 Slide 7 & $ % Notice that h ( y ) = y , and h 1 ( y ) = y . Then, D = { ( x, y ) IR 2 : h ( y ) = y x h 1 ( y ) = y, y y y 1 } . Z Z D f ( x, y ) dxdy = Z 1 " Z y y ( x 2 + y 2 ) dx # dy, = Z 1 1 3 x 3 | y y + y 2 x | y y dy, = Z 1 1 3 ( y 3 / 2- y 3 ) + y 2 ( y 1 / 2- y ) dy, = Z 1 1 3 y 3 / 2- 1 3 y 3 + y 5 / 2- y 3 dy, = 1 3 2 5 y 5 / 2 | 1- 1 3 1 4 y 4 | 1 + 2 7 y 7 / 2 | 1- 1 4 y 4 | 1 , = 2 15- 1 12 + 2 7- 1 4 = 9 3 5 7 . Slide 8 & $ % Find the R R D f ( x, y ) dxdy for f ( x, y ) = 1 , D = ( x, y ) IR 2 : x 2 9 + y 2 4 1 ....
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w8-C - Math 20C Multivariable Calculus Lecture 21 1 Slide 1...

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