数学分析 高等数学 微积分 英语课件 上海交通大学Chapter5a

xi i / n,i 0,1, , n. Then take xi xi i / n as the sample points. By taking limit to the Riemann sum, we have
1 x2dx lim
0
n
n i1
f (xi )xi
lim n
n ( i )2 i1 n
2. If
sin xdx 2, find
0
sin sin 2
lim( n n n n 1 n 1
2
sin n
n n 1
).
n
n n n
n
n
1 sin xdx 2 .
0
Exercise
1. Express the limits into definite integrals:
(1)
lim
1
1 2
(1 en
2 2
en
( n 1) 2
e n ).
n n
(2)
lim(
n
2 Hale Waihona Puke Baidun2 12
2 4n2 22
2 ). 4n2 n2
Idea: first, divide the region S into n subregions by partitioning [a,b] into n subintervals [xi-1,xi] (i=1,L,n)
with x0=a and xn=b; then, approximate each subregion Si by a rectangle since f(x) does not change much and can be
can be treated as a constant; last, make sum di and take
n
i 1
limit time
ilnnimteriv1ald[i a, ,bi]f
the
is d
limit exists, n
lim
n
i 1
di.
then
the
distance
in
the
1
n
1
1
n 1 f ( i ),
n1 n2
2n i1 n 1 i i1 n n
with f (x) 1 . Therefore,
n
1 x
lim( 1 1
n n 1 n 2
1 ) 1 1 dx. 2n 0 1 x
The other solution is 2 1 dx. 1x
Example
Example
Ex. Find 1 x2dx by definition of definite integral. 0
Sol. To evaluate the definite integral, we partition [0,1]
into n equally spaced subintervals with the nodal points
fxii stihneteRgireambalnenosnu[ma,bh]asanlidmIitislitmh0ei1def f(ixni )itexiintIe,gtrhaelnowf ef call
b
from a to b, which is denoted by I f (x)dx. a
Remark
Ex. If sin xdx 2, find the limit 0
lim 1 (sin sin 2 sin (n 1) ).
n n
n
n
n
Sol. lim 1 (sin sin 2 sin (n 1) )
n n
n
n
n
1 lim (sin sin 2 sin n )
Idea: first, divide the time interval [a,b] into n subintervals;
then, approximate the distance di in each subinterval [ti-1,ti]
by di¼(ti-ti-1)v(xi) since v(t) does not vary toonmuch and
the
region
has
area
lim
n
i1
Si.
Remark
In the above limit expression, there are two places of
significant randomness compared to the normal limit
expression: the first is that the nodal points {xi} are
1 n
lim
n
n(n
1)(2n 6n3
1)
1. 3
Example
11
1
Ex. Express the limit lim( ) into a
definite integral. n n 1 n 2
2n
Sol. Since 1 1 1 , we have
ni n 1 i
1
1
n
b
In the notation f (x)dx, a and b are called the limits of integration; a ias the lower limit and b is the upper limit;
f(x) is called the integrand.
treated as a constant in each subinterval [xi-1,xi], that is,
Ssui¼m(xi-nxiS-1i)fa(nxid),twakheelriemxiit
i 1
n
islnimanyin1pSoii,nitfinth[exli-i1m,xiit];exlaisstt,s,mthaekne
b
The definite integral f (x)dx is a number; it does not depend on x, that is, wae can use any letter in place of x:
b
b
b
a f (x)dx a f (t)dt a f (r)dr.
arbitrarily chosen, the second is that the sample points {xi}
are arbitrarily taken too. n
S
lim
n
i1
Si
means, no matter how {xi} and {xi} are
n
chosen,
the
limit
differentiation calculus. The area and distance problems are two typical
applications to introduce the definite integrals
The area problem
Problem: find the area of the region S with curved sides, which is bounded by x-axis, x=a, x=b and the curve y=f(x).
If f takes on both positive and negative values, then the b integral f (x)dx is the net area, that is, the algebraic sum of araeas
The distance traveled by an object with velocity v=v(t), b during the time period [a,b], is v(t)dt. a
lim
n
i1
Si
always exists and has same value.
The distance problem
Problem: find the distance traveled by an object during the time period [a,b], given the velocity function v=v(t).
Ex. Use the definition of definite integral to prove that b f (x) c is integrable on [a,b], and find cdx. a
Interpretation of definite integral
b
If f (x) 0, the integral f (x)dx is the area under the a curve y=f(x) from a to b
Definition of definite integral
We call p : a x0 x1 xn1 xn b a partition of the interval [a,b]. m1iaxn {xi} is called the size of the partition, where xi xi xi1n(i 1, , n). xi [xi1, xi ](i 1, , n) are called
Furthermore, the sample points are usually chosen by xi xi1 or xi xi thus the Riemann sum is given by
n1 b a f (a i(b a)) or n b a f (a i(b a))
The usual way of partition is the equally-spaced partition xi a ih, i 0,1, n; h (b a) / n
so the size of partition is h (b a) / n
In this case 0 is equivalent to n
Introduction to integrals
Integral, like limit and derivative, is another important concept in calculus
Integral is the inverse of differentiation in some sense There is a connection between integral calculus and
sample points. f (xi )xi is called Riemann sum.
Definition Supip1ose f is defined on [a,b]. If there exists a
constant I such that for any partition p and any sample points n
i0 n
n
i1 n
n
Example
Ex. Determine a region whose area is equal to the given
limit
(1) lim
n
2 (5 2i )10
n n i1
n
n i
(2) lim
tan
n i1 4n 4n
Definition of definite integral