Transcript 定積分面積與黎曼和

第二單元
面積與黎曼和
Area
b
b
r
h
a
a ×b
a
1
a ×h
2
h
a
pr
2
1
(a + b) ×h
2
Archimedes’ Method
287 B.C. ~ 212 B.C.
P1
P2
Area of a circle = lim( Pn )
n® ¥
P3
Area
More Rectangle is better
n
A»
å
i= 1
f ( xi )D x
The Area under the Curve
Definition
f defined on[a,b],if f is continuous on[a,b]and
f ( x) ³ 0 on[a,b], then
n
A = lim An = lim å f ( xi )D x
n® ¥
n® ¥
i= 1
Riemann Sum
 A German mathematician who made important generalization to the
definition of the integral(1826-1866)
The Definite integral

ò
b
a
n
f ( x)dx = lim å f ( xi )D x
n® ¥
i= 1
f defined on [a,b]for which the limit exists and
is same for any choice of partitions
.When the limit exists , we say that f is integrable
on [a,b].
Some Special Sum Formula
n
(1)
å
i = 1+ 2 + 3 + 4 + L L L + n
i= 1
n
(2)
å
i= 1
n( n + 1)
=
2
i 2 = 12 + 22 + 32 + 42 + L L L + n2
n(n + 1)(2n + 1)
=
6
n
3
(3) å i = 13 + 23 + 33 + 43 + L L L + n3
i= 1
2
én(n + 1) ù
ú
=ê
êë 2 ú
û
Example
Write the indicated sum in sigma notation
56
(1) 1+ 2 + 3 + L L + 56 =
å
k=1
(2) 1+ 1/ 2 + 1/ 3 + L L + 1/100 =
k
100
1
k
å
k=1
100
(3) 1- 1/ 2 + 1/ 3 + L L - 1/100 =
å
k+1
(- 1)
k=1
(4) f (a1 )D x + f (a2 )D x + L L + f (an )D x =
n
å
i= 1
f (ai ) D x
1
k
Example5
Find
å
(3k + 2)
k= 1
Solution :
5
å
k= 1
=3
5
(3k + 2) =
5
å
k= 1
å
k= 1
5
3k +
å
2
k= 1
k + (2 + 2 + 2 + 2 + 2)
5 ×6
+ 10 = 45 + 10 = 55
= 3×
2
Example
12 + 22 + 32 + 42 + L L L + 102 = ?
5 10 ×11 ×21 7
6 2
= 5 ×11×7 = 385
n
(2)
å
i 2 = 12 + 22 + 32 + 42 + L L L + n2
i= 1
n(n + 1)(2n + 1)
=
6
Example
Compute
ò
1
2
x dx
0
1
Dx =
n
Height :
1
2
x1 = , x2 = , L
n
n
i
n
, xi = , xn =
n
n
1 2
2 2
f ( x1 ) = ( ) , f ( x2 ) = ( ) , L L
n
n
i 2
Height : f ( xi ) = ( )
n
Example
the area of the ith rectangle is
1 i 2 i2
( ) ×( ) = 3
n n
n
n
Example
An =
å
i= 1
2
æi ÷
ö
f ( xi )D x = å çç ÷ D x
÷
ç
è
ø
n
i= 1
n
2
n
æi ö
æ
ö
1
1
2
÷
ç
=
i
= å çç ÷
÷
÷
ç
÷
÷ n3 åi= 1
ç
ç
è
ø
è
ø
n
n
i= 1
n
1 n(n + 1)(2n + 1)
= 3×
n
6
n(n + 1)(2n + 1) 1
A = lim
=
3
n® ¥
6n
3
\
ò
1
0
1
x dx =
3
2
單元結語
 本單元說明了定積分的意義為面積。
 面積的求法要能夠以黎曼和求之，黎曼和就是分割後再相
加的觀念。
 以後是不是要求面積時都需要以黎曼合來計算呢?不！這
樣算實在太麻煩了，第四單元，有一個定理叫做”微積分
基本定理”，它可簡化我們的計算。
 看完本單元後單元後，建議同學多複習上學期所學的各類
型導函數的求法。