Page 1 | © 2012 by Janice L. Epstein 6.2 □ Area Area (Section 6.2)
EXAMPLE 1
What is the area under the curve y = -2 x + 2 from 0 to 1?
Page 2 | © 2012 by Janice L. Epstein 6.2 □ Area 2
1
Approximate with rectangles:
2
1
1
1
More generally, partition the interval [ a, b ] as
P = {a = x0 , x1 ,, xn = b}
A subinterval [ xi-1 , xi ] has a width Dxi = xi - xi-1 and the point xi*
is in that subinterval.
2
The length of the longest subinterval is the norm of P,
P = max {Dx0 , Dx1 ,, Dxn }
The area of the ith rectangle is
Ai = f ( xi* )Dxi .
1
The total approximate area is the sum
1
n
å f ( x )Dx
*
i
i
i=1
n
We say the area of the region is A = lim å f ( xi* )Dxi
P 0
i=1
Page 3 | © 2012 by Janice L. Epstein 6.2 □ Area EXAMPLE 2
You are given a function f, an interval, partition points, and a
description of the point xi* within the ith subinterval.
(a) Find P
(b) Sketch the graph of f and the approximating rectangles.
(c) Find the sum of the approximating rectangles.
(i) f ( x ) = 16 - x 2 , [ 0, 4], P = {0, 1, 2, 3, 4} , xi* = left endpoints.
é pù
ì p p p pü
(ii) f ( x ) = 4cos x, ê 0, ú , P = ïí0, , , , ïý , xi* = right endpt
ï
ïî 6 4 3 2 ï
ïþ
ëê 2 ûú
(iii) f ( x ) = ln x and interval [1, 3]. Create the partition by using
3 subintervals of equal length and xi* = midpoints.
Page 4 | © 2012 by Janice L. Epstein 6.2 □ Area Page 5 | © 2012 by Janice L. Epstein 6.2 □ Area EXAMPLE 3
For the following functions, set up the area as a limit of a sum. Do
not evaluate the limit.
(i) f ( x ) = x 2 + 3 x - 2 on the interval [1, 4] using right endpoints.
(ii) f ( x) = x 2 + 1 on the interval [0, 5] using left endpoints.
Page 6 | © 2012 by Janice L. Epstein 6.2 □ Area EXAMPLE 4
The following limits represent the area under the graph of f ( x )
from x = a to x = b . Identify f ( x ), a and b.
æ 3 ö÷ n
3i
(i) lim çç ÷÷ å 1 +
n¥ ç
è n ø i=1
n
n
æ10 ö
1
(ii) lim çç ÷÷÷ å
3
n¥ ç
è n ø i=1
æ
10 ö÷
ç
1 + ç7 + i÷÷
çè
n ø