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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 double integral on rectangles. Average of a function. Double integrals general domains (Sec. 15.2). Examples of double integrals. Slide 2 & \$ % Integral of a single variable function Definition 1 (Integral of single variable functions) 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 & \$ % Double integrals on rectangles Definition 2 (Double integrals on rectangles) 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 we have introduced a partition on R by rectangles of side Δ x = ( x 1 - x 0 ) /n , Δ y = ( y 1 - y 0 ) /n . We denoted
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w7-C - Math 20C Multivariable Calculus Lecture 19 1 \$...

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