The 1st or x-coordinate evaluates to the the right endpoint of each subinterval. For example, the 1st subdivision is 0<=t<=1/n ---> 0<=nt<=1 ---> ceil(nt)=1 ---> (1/n)ceil(nt) = 1/n, the x-value at the right endpt of the first subinterval. The key is that the ceil( ) function replaces ALL t-values in that subinterval by a constant!
23
The 2nd or y-coordinate in the parametrically defined function is more sophisticated. Its values range from 0 to f(right endpt). Try it for n=3 and some subdivision like 0<=t<=1/3 or, better, 1/3<=t<=2/3.
24
less than or equal to "t" less than or equal to
25
1
50
26
The summation below evaluates the right Riemann sum for n rectangles. The factor "1/n" represents the base of each rectangle and is constant for each value of n. Therefore you can take it outside the summation. Try it!
27
"f(k/n)" is the height at the right endpt of each subdivision. Therefore the product (1/n) f(k/n) is the area of each approximating rectangle.
28
equals
29
30
31
powered by
powered by
New Blank Graph
Examples