Question 1031352: Function f(x) is positive, increasing and concave down on the closed interval [a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the left sum, right sum, and trapezoidal rule approximations for the value of the integral from a to b of f of x dx Which one of the following statements is true?
Trapezoidal rule value < Left sum < Right sum
Left sum < Trapezoidal rule value < Right sum
Right sum < Trapezoidal rule value < Left sum
Cannot be determined without the x‐values for the partitions
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Left sum < Trapezoidal rule value < Right sum
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For the function function f(x) that is positive, increasing and concave down on the closed interval [a, b],
The left sum (or lower Riemann sum, w/c makes use of the left endpoint of the subinterval) is less than the right sum (or upper Riemann sum, w/c makes use of the right endpoint of the subinterval). The trapezoidal approximation makes use of both endpoints, but has an area that is the average of the areas of the inscribed and circumscribed rectangles. (Verify!)
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