Using rectangles of unit width on the interval [3,5], arrange the curves in ascending order of the estimated areas under them.

Your other posting of this question is answered here: link

Numbered 1–6 from top to bottom, functions 1, 2, 6, 4, 5, 3 are in order by decreasing area under the curve.

There are several ways to use rectangles of unit width. The above-referenced answer uses the midpoint of each rectangle. Since it is unit width, the function value at that point is the area of the rectangle. Adding f(3.5) and f(4.5) gives an approximation of the area under the curve according to the midpoint rule.

Several of the functions are quadratic. Their area can be found exactly using Simpson's rule. That rule adds 4 times the function value at the midpoint of the interval (4×f(4)) to the sum of function values at the ends, f(3) + f(5). The total is divided by 3 to get the approximate area.

You can also use left-end values: f(3) + f(4). Or you can use right-end values: f(4) + f(5).

These curves are flat enough that the left-, right-, or mid-point rules should work well enough to put the areas in proper order. A trapezoidal rule can also be used. That formula would be area = (f(3) + 2f(4) + f(5))/2.

Note that the rule we used only requires the functions be evaluated twice. The left- and right- rules also are that simple to compute. The trick is to choose a method likely to be "good enough" before you do a lot of computation.

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