MTH 252 Integral Calculus Chapter 6 – Integration Section 6.4 – The Definition of Area as a Limit; Sigma Notation Copyright © 2005 by Ron Wallace, all rights reserved.

The Area Problem (from 6.1)  Given a continuous non-negative function f(x) over an interval [a, b]; find the area bounded by f(x), the x-axis, x=a, & x=b. x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x0x0 x7x7 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7

The Area Problem (from 6.1) x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x0x0 x7x7 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7 Some type of shorthand notation would be helpful with these types of expressions.

Sigma Notation Example: Each term of the sum is the function: In the sum, k takes on ALL integral values from 3 through 8. So the expression is shortened as follows:

Sigma Notation Example: Each term of the sum is the function: In the sum, k takes on ALL integral values from 1 through 6. So the expression is shortened as follows: Same problem, alternate solution!

Sigma Notation NOTE: m and n MUST be integers with m ≤ n. “lower limit” “upper limit” “index”

Sigma Notation Changing the Limits Previous Example: ? Note that both limits were DECREASED by 2 and in the function k was replaced by k+2. Sometimes the letter for the index is changed, but this is not necessary!

Sigma Notation Changing the Limits In General: Note that both limits are DECREASED by h and in the function k is replaced by k+h. Sometimes the letter for the index is changed, but this is not necessary! h is any integer (positive or negative).

Sigma Notation One more note on notation. Instead of using functional notation, subscripted notation is often used. That is, …

Properties of Sums c is any expression not dependent on k.

Useful Summation Formulas Prove by Induction!

Back to the Area Problem  Given a continuous non-negative function f(x) over an interval [a, b]; find the area bounded by f(x), the x-axis, x=a, & x=b. x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x0x0 x7x7 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7

The Area Problem Refined x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x0x0 x7x7 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7 If the x ’s are equally spaced, then … The heights of the rectangles can use any value of x in the interval:

The Area Problem Summarized

Example Determine the area under f(x)=4-x 2 over [0, 2].