Riemann Sum vs Simpson's Rule

[jungykang]

Hey guys,

While studying some different forms of integration, i came across Riemann sums and Simpson's rule.
I was wondering what the biggest difference between the two are and the differences between their uses are.

Thanks

 

[stapel]

While studying some different forms of integration, i came across Riemann sums and Simpson's rule.
I was wondering what the biggest difference between the two are and the differences between their uses are.

That really sounds like a homework exercise.... What are your thoughts so far on this question? For instance, when you applied each of the algorithms to one particular problem, what differences did you notice?

 

[HallsofIvy]

"Riemann sums", which are really used to define the integral and while can be used as a numerical method of integration but seldom is, approximates the function to be integrated by a "piecewise constant" function while Simpson's rule approximates it by a "piecewise quadratic" function. One can show that the error for "Riemann sums" varies as $\displaystyle \Delta x$ \(\displaystyle \ while \ the \ error \ for \ "Simpson's rule" \ varies \ as \ \left(\Delta x\right)^3.\text That \ is, \ for \ the \ same \ size \ "step", and approximately the same amount of work, Simpson's rule gives much better accuracy.

(You don't mention it but the "Trapezoid rule" is midway between these. It approximates the function by a "piecewise linear" function and its accuracy varies as\) $\displaystyle \left(\Delta x\right)^2$.)

 

[jungykang]

That really sounds like a homework exercise.... What are your thoughts so far on this question? For instance, when you applied each of the algorithms to one particular problem, what differences did you notice?

Well the first thing I've noticed is that the Riemann sums do not require certain set intervals, while the simpson's method does and from this you can definately infer different usages. I'd say you could use the Simpson's rule for integrating values when say a sensor gives the flowrate of blood in certain intervals (which is the topic I am exploring), while the Riemann sum tends to be more flexible.

 

[jungykang]

"Riemann sums", which are really used to define the integral and while can be used as a numerical method of integration but seldom is, approximates the function to be integrated by a "piecewise constant" function while Simpson's rule approximates it by a "piecewise quadratic" function. One can show that the error for "Riemann sums" varies as $\displaystyle \Delta x$ \(\displaystyle \ while \ the \ error \ for \ "Simpson's rule" \ varies \ as \ \left(\Delta x\right)^3.\text That \ is, \ for \ the \ same \ size \ "step", and approximately the same amount of work, Simpson's rule gives much better accuracy.

(You don't mention it but the "Trapezoid rule" is midway between these. It approximates the function by a "piecewise linear" function and its accuracy varies as\) $\displaystyle \left(\Delta x\right)^2$.)

This was a brilliant explanation, thank you so much! You seem to be so active in these forums