"left" rule: 1135 ft"right" rule: 1215 ft
Step-by-step explanation:
The business of figuring the area under a curve that is defined by numerical values at different points on the curve is called "numerical integration." There are a number of ways it can be done.
Left
One of the easiest ways is to figure the area of each rectangular section under the curve using the distance between points as the width of the rectangle and the function value at the left side of the rectangle as its height. In terms of table values, you multiply the difference between two sequential times by the speed corresponding to the left time. This product is the an estimate of the distance traveled in that time interval. The sum of these products is the total distance traveled over the whole time period.
Since all the time intervals are the same width (5 seconds), we can simply add the first 6 speeds and multiply that sum by 5 seconds. Doing so gives ...
25 +31 +35 +43 +47 +46 = 227
(227 ft/s)×(5 s) = 1135 ft
The graph in the first attachment shows the rectangles whose area we summed.
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Right
Similarly easy is using the speed value at the right end of each time interval. Summing those, we get 243 ft/s, so the estimated distance is ...
31 +35 +43 +47 +46 +41 = 243*
(243 ft/s)×(5 s) = 1215 ft
The second attachment shows the rectangles whose area we summed for this estimate.
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Other Rules
The "trapezoidal rule" treats the area under the curve between the given points as though it were a trapezoid, not a rectangle. The net effect is the same as if we averaged the left- and right-rule results. Doing that would give a distance estimate of ...
(1135 ft +1215 ft)/2 = 1175 ft
The way we have drawn the graph, using straight lines between the points, the area under the (piecewise linear) curve would match this value.
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If we approximate the curve between the two points using a quadratic function, we arrive at "Simpson's Rule." This makes a further modification to the way we sum the data values. (It is explained in numerous videos and web sites, and is beyond the scope of this answer.) The result from that rule is an estimate of 1183 1/3 feet.
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* Note that we can obtain the "right" sum by starting with the "left" sum and adding the difference between the last and first data points:
227 +(41-25) = 243
