RIEMANN SUMS and AREA APPROXIMATIONS	   Write-pair-share

The five standard approximation methods for the area under a curve are:  

approximation method	abbreviation	
Riemann sum—left endpoints	L	
Riemann sum—midpoints	M	
Riemann sum—right endpoints	R	
Trapezoidal Rule	T	
Simpson’s Rule (parabolas)	S	

Your objective is to predict which approximation methods will provide the most accurate estimates of the area under the curve for each of the functions listed below (there may be methods that are equally accurate in some cases).  In order to accomplish this objective, it will be helpful to sketch graphs of each function and include some representative rectangles, trapezoids or parabolas in order to better visualize the applications of the area methods.  

In particular, please sketch five graphs for each function, one for each approximation method.  In each graph, divide the domain into four intervals of equal width and sketch four representative rectangles, trapezoids or parabolas (whichever is appropriate for the approximation method).  Afterwards, complete the table below:

For each function, please list the approximation methods in descending order, based on predicted accuracy.

Note:  For some functions, some of the methods may be equally accurate; please indicate these in your entries, e.g., T=S, L=M=R, etc.

function	domain	approximation methods
(descending order, based on predicted accuracy)	
f(x)=x	[0, 10]		
f(x)=x^2	[0, 10]		
f(x)=sin(x)	[0, 6.28318]		
f(x)=sin(x)	[0, 3.14159]		
f(x)=3	[0, 10]		
f(x)=10 –  x^2	[0, 10]