Question 1119378: The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of 2∫1 f(x) dx .
x 1 1.1 1.3 1.6 1.7 1.8 2.0
f(x) 1 3 5 8 10 11 14
Answer by greenestamps(13200)   (Show Source):
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For each trapezoidal section, the area is the width of the section (difference in x values) times the average height of the section; the average height of each section is the average of the function values at the beginning and end of the section.
(1) (1,1) to (1.1,3): width 1.1-1 = 0.1; average height (1+3)/2 = 2; area (0.1)(2) = 0.2
(2) (1.1,3) to (1.3,5): width 1.3-1.1 = 0.2; average height (3+5)/2 = 4; area (0.2)(4) = 0.8
(3) (1.3,5 TO (1.6,8): width 1.6-1.3 = 0.3; average height (5+8)/2 = 6.5; area (0.3)(6.5) = 1.95
(4) (1.6,8) to (1.7,10): width 1.7-1.6 = 0.1; average height (8+10)/2 = 9; area (0.1)(9) = 0.9
(5) (1.7,10) to (1.8,11): width 1.8-1.7 = 0.1; average height (10+11)/2 = 10.5; area (0.1)(10.5) = 1.05
(6) (1.8,11) to (2,14): width 2-1.8 = 0.2; average height (11+14)/2 = 12.5; area (0.2)(12.5) = 2.5
Total area estimate: 0.2+0.8+1.95+0.9+1.05+2.5 = 7.4
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