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# w7-C - Math 20C Multivariable Calculus Lecture 19 1 \$...

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Math 20C Multivariable Calculus Lecture 19 1 Slide 1 & \$ % Double integrals (Sec. 15.1 - 15.2) Review of the integral of single variable functions. Definition of a double integral on rectangles. Average of a function. Examples of double integrals in rectangles (sec. 15.2) Slide 2 & \$ % Integral of a single variable function Definition 1 Let f ( x ) be a function defined on a interval x [ a, b ] . The integral of f ( x ) in [ a, b ] is the number given by Z b a f ( x ) dx = lim n →∞ n X i =0 f ( x * i ) Δ x, if the limit exists. Given a natural number n we have introduced a partition on [ a, b ] given by Δ x = ( b - a ) /n . We denoted x * i = ( x i + x i - 1 ) / 2 , where x i = a + i Δ x , i = 0 , 1 , · · · , n . This choice of the sample point x * i is called midpoint rule.

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Math 20C Multivariable Calculus Lecture 19 2 Slide 3 & \$ % Riemann sum of a single variable function y y x 0 x 1 x 2 x 3 x 4 f(x) Slide 4 & \$ % Double integrals on rectangles Definition 2 Let f ( x, y ) be a function defined on a rectangle R = [ x 0 , x 1 ] × [ y 0 , y 1 ] . The integral of f ( x, y ) in R is the number given by Z Z R f ( x ) dxdy = lim n →∞ n X i =0 n X j =0 f ( x * i , y * j ) Δ x Δ y, if the limit exists. Given a natural number n , the partition on R are rectangles of side Δ x = ( x 1 - x 0 ) /n , Δ y = ( y 1 - y 0 ) /n . Let x * i = ( x i + x i - 1 ) / 2, y * j = ( y j + y j - 1 ) / 2, where x i = x 0 + i Δ x , and y j = y 0 + j Δ y , for i, j = 0 · · · , n . These sample points x * i , y * j are called midpoint rule.
Math 20C Multivariable Calculus Lecture 19 3 Slide 5 & \$ % Partition of the domain of a two variable function x y y y x x x x x y y 0 0 4 4 1 1 y 2 2 3 3 z Slide 6 & \$ % Double integrals of f ( x, y ) are volumes in IR 3 If f ( x, y ) 0, then R R R f ( x, y ) dxdy = V the volume above R and below the surface given by the graph of f ( x, y ). z x y f(x,y) R

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Math 20C Multivariable Calculus Lecture 19 4 Slide 7 & \$ % The order of integration can be switched in double integrals of continuous functions Theorem 1 (Fubini) If f ( x, y ) is a continuous function in R = [ x 0 , x 1 ] × [ y 0 , y 1 ] , then Z Z R f ( x, y ) dxdy = Z y 1 y 0 Z x 1 x 0 f ( x, y ) dx dy, = Z x 1 x 0 Z y 1 y 0 f ( x, y ) dy dx.
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w7-C - Math 20C Multivariable Calculus Lecture 19 1 \$...

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