Riemann Sums

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Riemann Sums

• Approximating Area

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One of the classical ways of thinking of an area
under a curve is to graph the function and then
approximate the area by drawing rectangular or
trapezoidal regions under the curve (or nearly
so).
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Finding the area of each region is easy with
rectangles, and these areas can be added to
approximate the area of the function.
Here we approximate the area under f between a
and b. Notice that the rectangles do not all have
the same width.
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This area is an approximation of
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This method is one of several that can be used if
we are unable to find an antiderivative of a
function, and therefore cannot evaluate an
integral symbolically, as we have been doing.
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This methd can be used when an antiderivative can
be found, but this is usually done for
instructional purposes.
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We start with a simple polynomial function
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In order to approximate the area under the curve
from 0 to 4, we can use a left sum with 2
subintervals. Note the way to write and
calulate L2.
The left sum uses the height at the left side of
each subinterval.
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Similarly we find a right sum with 2
subintervals. The right sum uses the height at
the right side of the subinterval.
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A midpoint sum uses the midpoint of each
subinterval to define the height.
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Finally, a trapezoidal sum uses the area of
trapezoids instead of reactangles. These may be
calculated from the function, or recognized as
the average of L2 and R2.
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If we increase the number of subintervals to 4 we
can repeat the processes.
The approximation will be better with smaller
subintervals.
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Similarly with the right sum.
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And for the midpoint sum.
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And the trapezoidal sum.
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As the number of subintervals increase the
approximation get closer to the true area under
the curve.
Usually, the calculations are not made by hand,
but utilize software for this purpose.
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You have probably noticed that sometimes you can
predict whether a given sum will be an over or
underestimate.
Talk this over and see if you can come up with
circumstances where you can predict over- or
underestimates.
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The heights are always determined by the value of
the function, sometimes given in a table,
sometimes found from a formula.
This is a very brief overview and should be
supplemented by reading Section 5.6 in your text.