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Lecture 13

MTH 162 Lecture 13: 6.5 Approximate Integration Notes
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Department
Mathematics
Course
MTH 162
Professor
Pachero
Semester
Fall

Description
MTH 162 Calculus II 6.5 Approximate Integration Notes L. Sterling Abstract Provide a generalization to each of the key terms listed in this section. Midpoint Rule What is the midpoint rule? ▯ The midpoint rule is used to approximate the area that’s under the function along with the x-axis thanks to subdividing the area into rectangles. Function Rb When it comes to the midpoint rule, to help approximate a f (x)dx, then it would be best to use the following where [a;b] is partitioned into n subintervals with of equivalent length, which is ▯ = b▯a: n ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ x0+ x 1 x1+ x 2 x n▯1+ x n ▯x f + f + ▯▯▯ + f 2 2 2 Example R3 2 By using 4 rectangles, use the Midpoint Rule to estimate1 x dx. ▯x = b▯a = 3▯1 = 2 = 1 n 4 4 2 3 5 a = 1; 2;2;2;3 = b ▯ ▯ ▯1+ 3▯ ▯ 5 ▯ ▯ ▯ ▯2 f x0+x1 = f 2 = f 2 = f 5 = 5 = 25 = 1:5625 ▯ 2 ▯ ▯32 ▯ ▯ 7 ▯ ▯ ▯ ▯ 16 x1+x2 2+2 2 7 7 2 49 f 2 = f ▯ 2 ▯ = f ▯ ▯ = f 4 = 4 = 16 = 3:0625 ▯x2+x3▯ 2+ 2 2 ▯9▯ ▯9▯2 81 f 2 = f 2 = f 2 = f 4 = 4 = 16 = 5:6025 ▯ ▯ ▯ 5 ▯ ▯ 11▯ ▯ ▯ ▯ ▯2 f x3+x4 = f 2+3 = f 2 = f 11 = 11 = 121 = 7:5625 2 2 2 4 4 16 1 1 2(1:5625 + 3:0625 + 5:0625 + 7:5625) =2 (17:25) = 8:625 Trapezoidal Rule What is the trapezoidal rule? ▯ The trapezoidal rule is used to approximate the area that’s under the function along with the x-axis thanks to subdividing the area into trapezoids. 1 Function R b When it comes to the trapezoidal rule, to help approximate a f (x)dx, then it would be best to use the following where [a;b] is partitioned into n subintervals with of equivalent length, which is ▯ = b▯a: n ▯(y +02y + 1y + 2▯▯ + 2y n▯1 + yn) T = 2 Then, the following would note that the L ns the the Riemann sums
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