RIEMANN SUMS
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A Riemann Sum is an approximation of the area under the curve for a function. It is found by
calculating the sum of the areas of rectangles used to estimate this area on a given interval. This
area is also known as an integral. Riemann sums can be expressed using left endpoints, right
endpoints, midpoints, or even random points on the curve known as sample points.
The integral of a curve does not always have to represent area. For instance, by integrating a
velocity function on a certain interval you are finding total distance traveled. The integral of a
function representing the rate at which snow accumulates would give you the total amount of
snow on the ground.
As stated before a Riemann Sum is an estimation of the area under the curve y = f(x) found by
approximating that space using rectangles. Rectangles above the x-axis signify positive area,
while rectangles below it represent negative area.
The bases of these rectangles (∆x) are found by dividing the interval on which you are
approximating the integral
, into the given amount of subintervals (n).
Base =
on the interval
The height of the rectangles is always a point on the function and depends on which type of
Riemann Sum is being used. For instance, if you were using right endpoints to find the sum, then
you would use the point furthest to the right in each subinterval
. The y-value for this point
would be the height of the rectangle on that subinterval.
Height =
The product of the base and this height would give you the approximation of the area under the
curve for just that subinterval.
Therefore the total Riemann Sum would be the sum of all the approximating rectangles from
each subinterval. The equation this right hand Riemann Sum can be written as:
Where
on the interval
Sigma notation is often used to write these sums more efficiently. For example:
Where
Therefore using sigma notation the left endpoint, right endpoint, midpoint, and sample point
Riemann sums can be written as:
Where
Where
is any number in the ith subinterval
The Trapezoidal Rule and Simpson’s Rule more accurately estimate the integral of function. The
Trapezoidal Rule uses trapezoids rather than rectangles to
approximate the area much more accurately.
Simpson’s Rule uses parabolas to approximate the integral even better than the Trapezoidal Rule
can.
The Riemann integral of a function, also known as a definite integral, can be found by
calculating the limit as the number of these approximating rectangles approaches infinity giving
a definite integral rather than just an approximation. As shown by the equation:
Examples
Ex. 1) Approximate the integral of
Riemann Sum.
Ex. 2) Approximate the integral of
Sum.
on the interval
on the interval
using the left endpoint
using the midpoint Riemann
Ex. 3) Evaluate
using the Riemann integral
Click here for an example of the Trapezoidal Rule
Click here for an example of Simpson’s Rule
Applications
Since Riemann sums and the Riemann integral are both ways of integrating, they can be
used almost anywhere you are calculating or approximating an integral. The most common
application of Riemann sums is in approximating area, usually where a function cannot be made
to represent the data being integrated in order to find the area underneath. For example, if you
were making a patio along your house with curved edges that were not easily represented by a
function, you could use Riemann sums to estimate the area of the patio in order to buy the right
amount of gravel to lay out for the base. They also have applications in exports and population
growth. The Riemann integral or definite integral can also be used to calculate the areas of
surfaces of revolution, where a curve is rotated around an axis.
History
Both Riemann sums and the Riemann integral were developed and named after the
German Mathematician, Bernhard Riemann. Georg Friedrich Bernhard Riemann (1826-1866)
studied under Carl Gauss at the University of Gottingen and eventually taught there himself. He
showed great skill in mathematics that was noted by many including Gauss who described his
“creative, active, truly mathematical mind and gloriously fertile originality”. Thanks to his work
integration was first “rigorously formalized”. His Riemann integral, that uses limits and his
Riemann sums, is still used today to define a definite integral. In addition to Integration, he made
advances in the theory of functions of a complex variable, mathematical physics, number theory,
the foundations of geometry and the basis for Einstein’s theory of relativity. Most important, due
to Riemann we now have an extremely confusing way of merely approximating an integral.
http://en.wikipedia.org/wiki/List_of_topics_named_after_Bernhard_Riemann
Profile
http://science.kennesaw.edu/~plaval/applets/Riemann.html
http://www.cut-the-knot.org/Curriculum/Calculus/RiemannSums.shtml
http://mathworld.wolfram.com/RiemannSum.html
http://en.wikipedia.org/wiki/Riemann_sum
http://archives.math.utk.edu/visual.calculus/4/riemann_sums.4/
http://www.math.hmc.edu/calculus/tutorials/riemann_sums/
http://calculusapplets.com/riemann.html
http://www.msstate.edu/dept/abelc/math/integrals.html
http://www.analyzemath.com/calculus/RiemannSums/RiemannSums.html
http://en.wikipedia.org/wiki/Riemann_integral
http://archives.math.utk.edu/visual.calculus/4/definite.1/index.html
http://www.intmath.com/Integration/5_Trapezoidal-rule.php
http://www.intmath.com/Integration/6_Simpsons-rule.php
http://integrals.wolfram.com/index.jsp
Videos
http://www.teachertube.com/viewVideo.php?video_id=71388&title=Riemann_Sums&vpkey=
http://www.youtube.com/watch?v=gFpHHTxsDkI
http://www.youtube.com/watch?v=GE4OLfmJ8P8
http://www.youtube.com/watch?v=xdXW-V0Q7xA
http://www.youtube.com/watch?v=t48qM0vEXJE&annotation_id=annotation_927406&feature=
iv
Applications
http://www.usna.edu/MathDept/website/courses/calc_labs/area/Application.html
http://curvebank.calstatela.edu/arearev/arearev.htm
http://upload.wikimedia.org/wikipedia/commons/1/19/Riemann_sum_%28leftbox%29.gif
http://upload.wikimedia.org/wikipedia/commons/6/61/Riemann_sum_%28rightbox%29.gif
http://upload.wikimedia.org/wikipedia/commons/c/c3/Riemann_sum_%28middlebox%29.gif
http://upload.wikimedia.org/wikipedia/commons/e/ee/Riemann.gif
http://www.cut-the-knot.org/Curriculum/Calculus/RiemannSums.shtml
http://archives.math.utk.edu/visual.calculus/4/definite.2/index.html