Expert: Paul Klarreich - 12/4/2009

Question
I am confused on this problem:
Consider f(x)=X^2 on [0,2] and a particular P given by P= {0, 2/n, 4/n,..., 2-2/n,2} Find U(P,f) and L(P,f) and use the definition to evaluate the intergral of x^2 from 0 to 2.

I know the partition is 2/n so you will eventually have to divide by that. It also envolves the left and right end points.  

Answer
Questioner: Sally
Country: United States
Category: Advanced Math
Private: No
Subject: U(P,f) and L(P,f)
Question: I am confused on this problem:
Consider f(x)=X^2 on [0,2] and a particular P given by P= {0, 2/n, 4/n,..., 2-2/n,2} Find U(P,f) and L(P,f) and use the definition to evaluate the intergral of x^2 from 0 to 2.

I know the partition is 2/n so you will eventually have to divide by that. (NO, I THINK YOU MULTIPLY BY THAT) It also envolves the left and right end points.
...........................................................

Here is the Riemann sum you must compute:

SUM(for k = 1 to n) f(chosen-x[k]) dx

The 'chosen-x[k]' depends on

1) The function.
2) The type of sum.

In this case, f(x) = x^2 is increasing on [0,2], so

The UPPER-sum-chosen-x[k] will be at the right side of the interval, so it will be  k * dx

The LOWER-sum-chosen-x[k] will be at the left side of the interval, so it will be  (k-1) * dx[k]

Now the  dx is  2/n -- you knew that.

So we are ready to go:

I'll do UPPER, you'll do LOWER.

SUM(for k = 1 to n) f(k * 2/n) 2/n

SUM(for k = 1 to n) (2k/n)^2 2/n

SUM(for k = 1 to n) 8k^2/n^3

8/n^3 SUM(for k = 1 to n) k^2

Now you look up:
                         n(n + 1)(2n + 1)
SUM(for k = 1 to n) k^2 = ----------------
                                6

and put that in:

  8   n(n + 1)(2n + 1)
= ---- -----------------
 n^3        6


and finally take  lim[n -> inf] of that, and get 8/3, the correct integral.

When you do the LOWER, you will have

                             (n-1)(n)(2n - 1)
SUM(for k = 1 to n) (k-1)^2 = ----------------
                                   6

which will give the same limit.  Try it.


IN THE FUTURE, be sure to define all the terms you use.  DO NOT just write  U(...) and assume I am your teacher and use the same notation.