Problem:  Use left-hand endpoint, right-hand endpoint, and midpoint Riemann sums to estimate the area under the graph of y = f(x) =  6/(x² + 1) from x = 1 to x = 5.  In other words, estimate f(x)dx.  The instructions below show how to use a TI-89 calculator to do this.  Of course, you still need to be able to write out such Riemann sums with paper and pencil.

 Estimates for n = 4

 Use F4 (Other) key; select Define

 Use F3 (Calc) key; select Σ (sum)

 Right-hand endpoint Riemann sum:

 Left-hand endpoint Riemann sum:

 You could use one very long expression like the following…
Σ(f(1.+i*(5.-1.)/4)*(5.-1.)/4,i,1,4)  
… but that's a bad idea!      

 Midpoint Riemann sum:

 Estimates for n = 10

 Estimates for n = 100

 Estimates for n = 1000

 (It will take about 4 minutes for each of these results for n = 1000.)

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