CK-12-Calculus

http://www.ck12.org Chapter 4. Integration

Definition

Letfbe a continuous function on a closed interval[a,b].LetPbe a partition ofnequal sub intervals over

[a,b].Then the area under the curve offis the limit of the upper and lower sums, that is

A=n→lim+∞S(P) =n→lim+∞T(P).

Example 3:
Use the limit definition of area to find the area under the functionf(x) = 4 −xfrom 1 tox= 3.

Solution:
If we partition the interval[ 1 , 3 ]intonequal sub-intervals, then each sub-interval will have length^3 −n^1 =^2 nand height
3 −i 4 xasivaries from 1 ton.So we have 4 x=^2 nand

S(P) =

n

∑ 1 (^3 −i^4 x)^4 x=

n

∑ 1 (^34 x)−

n

∑ 1 i(^4 x)^2

= ( 34 x)n−n(n 2 +^1 )( 4 x)^2.

Since 4 x=^2 n, we then have by substitution

( 34 x)n−n(n 2 +^1 )( 4 x)^2 = 6 −( 2 +^2 n)= 4 +^2 n→4 asn→∞. Hence the area isA= 4.
This example may also be solved with simple geometry. It is left to the reader to confirm that the two methods yield
the same area.

Lesson Summary

1. We used sigma notation to evaluate sums of rectangular areas.


2. We found limits of upper and lower sums.


3. We used the limit definition of area to solve problems.