consider the function f(x)= x^2/4 -6
Rn is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval.
Calculate Rn for f(x)= x^2/4 -6 on the interval [0,4] and write your answer as a function of n without any summation signs.
Rn= ?
after receving help on trying to figure out this problem i still find myself confused.
the Riemann sum of the first term was found to be 16/3 n(n+1/2)(n+1)/n^3 but i still don't understand what to do next. any help would again be appreciated-thanks.
The Riemann sum is:
Rn = 16/3 n(n+1/2)(n+1)/n^3 - 24
which is the requested answer. What you've done is to divide the graph in n rectangles. The upper right corner of the rectangle touches the graph of the function. Rn is the area of all the n rectangles summed together (negative height corresponds to negative area here). Clearly as n becomes larger and larger you obtain better and better approximations to the integral of the function.
If you take the limit n--> infinity of Rn you find that it is 16/3 - 24, which is indeed the integral of the function from zero to 4.
Also, for finite n, Rn is larger than its limiting value. This is because the function is increasing, causing the rectangles to overestimate the area inder the graph. This eror becomes smaller and smaller as you make n larger and larger.
ohhh thank you for your help!