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THE LIMIT DEFINITION OF A DEFINITE INTEGRAL


The following problems involve the limit definition of the definite integral of a continuous function of one variable on a closed, bounded interval. Begin with a continuous function $ y=f(x) $ on the interval $ [a, b] $. Let

$ a=x_{0}, x_{1}, x_{2}, x_{3}, $ ... $ , x_{n-2}, x_{n-1}, x_{n}=b $

be an arbitrary (randomly selected) partition of the interval $ [a, b] $ , which divides the interval into $ n $ subintervals (subdivisions). Let

$ c_{1}, c_{2}, c_{3}, $ ... $ , c_{n-2}, c_{n-1}, c_{n} $

be the sampling numbers (or sampling points) selected from the subintervals. That is,

$ c_{1} $ is in $ [x_{0}, x_{1}] $,

$ c_{2} $ is in $ [x_{1}, x_{2}] $,

$ c_{3} $ is in $ [x_{2}, x_{3}] $, ... ,

$ c_{n-2} $ is in $ [x_{n-3}, x_{n-2}] $,

$ c_{n-1} $ is in $ [x_{n-2}, x_{n-1}] $,

and

$ c_{n} $ is in $ [x_{n-1}, x_{n}] $ .

Define the mesh of the partition to be the length of the largest subinterval. That is, let

$ \Delta x_{i} = x_{i} - x_{i-1} \ \ $

for $ i = 1, 2, 3, ..., n $ and define

$ mesh = \displaystyle{ \max_{1 \le i \le n} \{ x_{i} - x_{i-1} \}} $ .

The definite integral of $ f $ on the interval $ [a, b] $ is most generally defined to be

$ \displaystyle{ \int^{b}_{a} f(x) \, dx}
= \displaystyle{ \lim_{mesh \to 0} \sum_{i=1}^{n} f(c_{i}) \Delta x_{i} } $ .

For convenience of computation, a special case of the above definition uses $ n $ subintervals of equal length and sampling points chosen to be the right-hand endpoints of the subintervals. Thus, each subinterval has length

equation (*) $ \ \ \ \ \ \ \ \ \Delta x_{i} = \displaystyle{ b-a \over n } $

for $ i = 1, 2, 3, ..., n $ and the right-hand endpoint formula is

equation (**) $ \ \ \ \ \ \ \ \ c_{i} = \displaystyle{ a + \Big( { b-a \over n } \Big) i } $

for $ i = 1, 2, 3, ..., n $ . The definite integral of $ f $ on the interval $ [a, b] $ can now be alternatively defined by

$ \displaystyle{ \int^{b}_{a} f(x) \, dx}
= \displaystyle{ \lim_{n \to \infty} \sum_{i=1}^{n} f(c_{i}) \Delta x_{i} } $ .

We will need the following well-known summation rules.

  1. $ \displaystyle{ \sum_{i=1}^{n} c = c + c + c + \cdots + c } $ (n times) $ = nc $ , where $ c $ is a constant
  2. $ \displaystyle{ \sum_{i=1}^{n} i = 1 + 2 + 3 + \cdots + n
= { n(n+1) \over 2 } } $
  3. $ \displaystyle{ \sum_{i=1}^{n} i^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2
= { n(n+1)(2n+1) \over 6 } } $
  4. $ \displaystyle{ \sum_{i=1}^{n} i^3 = 1^3 + 2^3 + 3^3 + \cdots + n^3
= { n^2(n+1)^2 \over 4 } } $
  5. $ \displaystyle{ \sum_{i=1}^{n} k f(i) }
= \displaystyle{ k \sum_{i=1}^{n} f(i) } $ , where $ k $ is a constant
  6. $ \displaystyle{ \sum_{i=1}^{n} (f(i) \pm g(i)) }
= \displaystyle{ \sum_{i=1}^{n} f(i) \pm \sum_{i=1}^{n} g(i) } $

Most of the following problems are average. A few are somewhat challenging. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by using the formulas given above in exactly the form that they are given. Solutions to the first eight problems will use equal-sized subintervals and right-hand endpoints as sampling points as shown in equations (*) and (**) above.





Click HERE to return to the original list of various types of calculus problems.


Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

kouba@math.ucdavis.edu



Duane Kouba 2000-06-08