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Home | 18.013A | Chapter 20 |
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From now on for our convenience we will consider Riemann sums in which all strips have the same width, .
A general Riemann sum with fixed widths consists then of the sum of multiplied by the values of chosen in each strip.
We define the following four choices that are of special interest.
If we always evaluate at the leftmost argument in each strip, call the value of the sum .
If we always evaluate at the rightmost argument in each strip, call the value of the sum .
If we evaluate at an argument that has maximum value for in each strip, call the sum .
If we evaluate at an argument that has minimum value for in each strip, call the sum .
We can make the following observations. We assume that is bounded so that and are both finite. Then the rightmost argument of one strip is the leftmost argument of the next.
Therefore the difference between and is only that gets a contribution of from the last strip that lacks and gets instead a contribution of that lacks.
The arguments in between at the endpoints of strips contribute to the interval to the right and an identical amount to the interval on the left.
Therefore we have
which implies that and come together as approaches 0.
Second, is greater than or equal to the true area and to any other Riemann sum for strips that can be obtained by subdividing the strips of width . , similarly is less than or equal to the true area or and to any other Riemann sum obtained by subdividing.
Finally, notice that if is increasing between and then and , which means that the true area and any Riemann sum obtained by subdividing gets sandwiched between bounds that come together as goes to 0.
Which means that all Riemann sums obtained by subdividing must approach the same value, and the function is integrable.
If
is not increasing between
and
but we can break up that interval into pieces in which
within each piece is increasing or decreasing, and
has bounded total variation between
and
, then
will be integrable by the same argument applied to all the pieces one at a time, and adding the resulting bound.
In the applet that follows, you can enter any standard function and limits and look at the the area computed by the left and right hand rules for different numbers of slices.
Exercises:
20.1 Compute for the function in the interval -1 to 2. If you add this to the left hand rule result for the integral of this function over this interval (get it from the applet below), what do you obtain? Try for different values.
20.2 Compute the left hand rule and right hand rule for this integrand using a spreadsheet, for and and .
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