A fun and interactive way of exploring how to compute definite integrals by calculating the area under a graph
using different variations of the Riemann Sum
Instructions
Below is a sample of what a basic riemann sum looks like between the interval of (-2, 2) with 20 partitions for
the function of 'x'.
1. To use this page click on the first expression and input
a
valid function with the format of 'f(x) = x' (THIS PAGE DOES NOT HANDLE MULTIVARIABLE
FUNCTIONS OR INVERSE FUNCTIONS.).
2. You will be able to use the sliders to control the interval that you
would
want to simulate. (Ensure that 'a' is less than 'b' or Riemann Sum will not be
calculated.)
3. Use the partitions to be able to increase of decrease the amount of partitions used to estimate
the
area under the curve.
4. Use the method to choose between a left point, midpoint, or right point sum method (Pick '0' for left point,
'0.5' for midpoint, and '1' for right point).
DO NOT CHANGE THE FORMULA ON EXPRESSION #7 AS THIS WILL CAUSE THE COMPUTATIONS TO FAIL.
Background Information on Riemann Sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after
nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of
functions or lines on a graph, but also the length of curves and other approximations.
The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics) that
together form a region that is similar to the region being measured, then calculating the area for each of these
shapes,
and finally adding all of these small areas together. This approach can be used to find a numerical approximation
for a
definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form
solution.
Because the region by the small shapes is usually not exactly the same shape as the region being measured, the
Riemann
sum will differ from the area being measured. This error can be reduced by dividing up the region more finely,
using
smaller and smaller shapes. As the shapes get smaller and smaller, the sum approaches the Riemann integral.