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第二节·微积分基本定理

Fundamentals of Advanced Mathematics (I)
Xueli Wang School of Science, BUPT Tel: 6228-2117 E-Mail: wangxlpku@https://www.doczj.com/doc/7015468307.html,
1

Section 5.2
The Fundamental Theorems of Calculus
2

Newton-Leibniz Formula
Definition. (Primitive function) If F ′( x ) = f ( x ) , x ∈ I , then F ( x ) is called an antiderivative of the function f ( x ) on I .
For instance, sin x is an antiderivative of cos x and ln x 1 . is an antiderivative of x The evaluation of a definite integral is closely related to the antiderivative of the integrand function.
3

Newton-Leibniz Formula
Example: Suppose that a particle moves along a straight line from t = a to t = b . If the velocity v = v ( t ) is known, then by the definition of definite integral we know that
s = ∫ v ( t )dt
a b
if the displacement function, s = s( t ) , is known, then
s = s( b ) ? s( a )
Hence, we have
∫
b a
v ( t )dt = s(b ) ? s(a )
It is well known that s′( t ) = v ( t ) or s( t ) is an antiderivative of v ( t ) , then, by the last equation, we can establish the following theorem.
4

Newton-Leibniz Formula
Theorem: (Newton-Leibniz formula) If antiderivative of f ( x ) on [a , b], then
f ∈ R[a , b] , and F ( x ) is an
b
∫
b a
f ( x )dx = F (b ) ? F (a ) = F ( x ) a = [ F ( x )]a
b
d
(
)
Newton, Isaac (1642-1727) English physicist and mathematician
Leibniz, Gottfried (1646-1716) German philosopher, physicist, and mathematician
5

Newton-Leibniz Formula
Proof: Arbitrarily insert n ? 1 points of partition between a and b such that Then,
a = x0 < x1 < " < xn?1 < xn = b
F (b ) ? F (a ) = [ F ( xn ) ? F ( xn?1 )] + [ F ( xn?1 ) ? F ( xn? 2 )] + " + [ F ( x2 ) ? F ( x1 )] + [ F ( x1 ) ? F ( x0 )] = ∑ [ F ( xk ) ? F ( xk ?1 )]
k =1 n
Since F ′( x ) = f ( x ) , so F ( x ) is continuous. By the mean value theorem of differential calculus gives
F ( xk ) ? F ( xk ?1 ) = F ′(ξ k )( xk ? xk ?1 ) = f ( ξ k ) Δxk
where xk ?1 ≤ ξ k ≤ xk and k = 1, 2,", n .
6

Newton-Leibniz Formula
Proof (continued) Since the antiderivative F ( x ) of f ( x ) exists and F ′( x ) = f ( x ) , then
F ( x ) is continuous, hence we may choose the
ξ k ∈ [ x k ?1 , x k ]
in the Riemann sum as the one obtained by the mean value theorem. Then
F ( b ) ? F (a ) = ∑ f (ξ k )Δxk
Taking limit on both sides we have
n d →0 k =1 k =1
n
F (b ) ? F (a ) = lim ∑ f (ξ k )Δxk = ∫ f ( x )dx
b a
Finish.
7

Newton-Leibniz Formula
Example Find the following definite integrals:
(1) (2)
∫
1 0
x 2 dx
∫
π
2 0
sin xdx
Solution:
1 1 3 2 = x dx = x (1) ∫0 3 0 3
1
1
(2)
∫
π
2 0
sin xdx = ? cos x 02 = 0 ? ( ?1) = 1
π
8

Newton-Leibniz Formula
Example: Find the value of
∫
?1 ?2
1 dx . x
1 , x < 0 , is ln | x | x
Solution: Since one of the antiderivative of
, then by Newton-Leibniz Formula, we have
∫
?1 ?2
?1 1 dx = [ ln | x |] ?2 x
= ln1 ? ln 2 = ? ln 2.
9

Fundamental theorems of Calculus
Definition: If f ∈ R[a , b], for any given x ∈ [a , b] , we get a definite value for the integral Hence, the definite integral
∫
x a
f ( t )dt .
∫
x a
f ( t )dt defines a
new function with respect to the upper limit x on the interval [a , b]. In general, if
f ∈ R[a , b] , then the integral
∫
x
a
f ( t )dt is called a integral with variable upper
limit x , and it is a function of x , where x ∈ [a , b] .
10

Fundamental theorems of Calculus
Note: If f ∈ R[a , b] and F ( x ) exists, by Newton-Leibniz theorem,
∫
x a
f ( t )dt = F ( x ) ? F (a )
where F ′( t ) = f ( t ) and hence
F ′( x ) = f ( x ).
By the differentiation with respect to x , we have
d x f ( t )dt = F ′( x ) = f ( x ). ∫ a dx
11

Fundamental theorems of Calculus
Theorem (The first fundamental theorem of calculus) If f ∈ C [a , b], then
x
Φ( x ) = ∫ f ( t )dt is differentiable on [a , b] and
a
Φ ′( x ) =
Proof Φ ( x + Δx ) = ∫
x +Δx a
d x f ( t )dt = f ( x ) . ∫ a dx
y
f ( t )dt
ΔΦ = Φ( x + Δx ) ? Φ( x )
=∫
x +Δx a
f ( t )dt ? ∫ f ( t )dt
a
x
Φ( x )
O
a
x
x + Δx
b
x
12

Fundamental theorems of Calculus
= ∫ f ( t )dt + ∫
a x x +Δx x
f ( t )dt ? ∫ f ( t )dt
a
x
=∫
x +Δx x
f ( t )dt ,
y
By the mean value theorem of integral calculus
ΔΦ = f (ξ )Δx ΔΦ = f (ξ ), Δx
ξ ∈ [ x , x + Δx ],
Φ(x)
ΔΦ lim = lim f (ξ ) Δx → 0 Δ x Δx → 0
o
a
x ξ x + Δx b
x
Δx → 0, ξ → x ∴ Φ′( x ) = f ( x ).
13

Fundamental theorems of Calculus
Theorem (The first fundamental theorem of calculus) If f ∈ C [a , b], then
Φ( x ) = ∫ f ( t )dt is differentiable on [a , b] and
a x
d x Φ ′( x ) = f ( t )dt = f ( x ) . ∫ a dx
Note: This theorem shows that for every continuous function f ( x ) there must exist the antiderivatives of f ( x ) , and the integral
∫
x a
f ( t )dt is just an
antiderivative of f ( x ) . This formula gives a connection between integral calculus and differential calculus. Hence, it is called the fundamental theorem of calculus.
14

Fundamental theorems of Calculus
Note: The formula means that the derivative of the integral with varying upper limit
∫
x
a
f ( t )dt with respect to the upper limit x is equal to the value
of the integrand at the upper limit x .
y M f ( x)
O
a
x
b
x
15

Fundamental theorems of Calculus
Note: Since a function is independent of the symbol used for its variable, f ( t )( t ∈ [a , b]) and f ( x )( x ∈ [a , b]) are exactly the same function. Hence, the formula in the last theorem sometimes may also be written as
d x f ( x )dx = f ( x ) ∫ a dx
But in this case, there may have some confusions. To avoid this, we usually change f ( x ) into f ( t ) , where a ≤ t ≤ x .
Note: We have seen that a new function any continuous function f ( x ) .
∫
x a
f ( t )dt has been well defined for
16

Fundamental theorems of Calculus
Example （P222） : Find the derivatives of the following functions ( x ≥ 1) : x sin t 1 sin t (1) Φ( x ) = ∫ dt ; (2) Φ( x ) = ∫ dt ; 1 x t t . x 2 sin t e2 x (3) Φ( x ) = ∫ dt ; (4) Φ( x ) = ∫ 2 ln tdt ; 1 x t
Solution.
d x sin t sin x (1) Φ ( x ) = dt = . ∫ 1 dx t x
'
d 1 sin t d ? x sin t ? sin x (2) Φ ( x ) = dt = ? dt = ? . ∫ ∫ ? ? x 1 dx t dx ? t x ? x 2 sin t (3) ∫ dt may be regarded as a com[osite function.Hence 1 t 2 2 x 2 sin t d d sin x sin x 2 Φ' ( x ) = dt ? x = ? 2x = 2 . ( ) 2 2 ∫1 t dx x x d(x )
'
17

Fundamental theorems of Calculus
Solution (continue): (4) taking many number c > 0,say,1,divide the integral into two parts. Then
e2 x d e2 x d ? 1 ? Φ ( x) = tdt = tdt + tdt ln ln ln ? ? ∫1 dx ∫ x 2 dx ? ∫ x 2 ? = ? ln x 2 ? 2 x + ln e 2 x ? 2e 2 x = 4 x (e 2 x ? ln x ). Finish. '
18

Fundamental theorems of Calculus
Example: Suppose that f ( x ) = ∫
Solution
1 dt , find f ′( x ) . sin x 1 + t 2
2
f ( x) = ? ∫
sin x
2
u 1 u = sin x 1 ?∫ 2 dt 2 dt 2 1+ t 1+ t
u d 1 ? ? du ′ f ( x) = d t ? ?? ? ∫2 2 du ? 1+ t ? dx
=?
? cos x 1 = ? cos x 1 + sin 2 x 1 + u2
19

Fundamental theorems of Calculus
Example: Find
∫ lim
x →0
1
cos x
e dt
?t2
x2
.
? 0? ? 0? ? ?
Solution
d 1 ?t2 d cos x ? t 2 e dt = ? ∫ e dt dx ∫cos x dx 1
= ?e
? cos 2 x
? (cos x )′
? cos 2 x
= sin x ? e
lim
x →0
∫
1
cos x
e ? t dt
2
x2
= lim
sin x ? e x →0 2x
? cos 2 x
=
1 . 2e
20

1-定积分与微积分基本定理(理)含答案版

定积分与微积分基本定理(理) 基础巩固强化 1.求曲线y ＝x 2与y ＝x 所围成图形的面积，其中正确的是( ) A ．S ＝?? ?0 1(x 2－x )d x B ．S ＝?? ?0 1 (x －x 2)d x C ．S ＝?? ?0 1 (y 2－y )d y D ．S ＝??? 1 (y － y )d y [答案] B [分析] 根据定积分的几何意义，确定积分上、下限和被积函数． [解析] 两函数图象的交点坐标是(0,0)，(1,1)，故积分上限是1，下限是0，由于在[0,1]上，x ≥x 2，故函数y ＝x 2与y ＝x 所围成图形的面积S ＝?? ?0 1 (x －x 2)d x . 2．如图，阴影部分面积等于( ) A ．2 3 B ．2－3 [答案] C [解析] 图中阴影部分面积为

S ＝??? -3 1 (3－x 2 －2x )d x ＝(3x －1 3x 3－x 2)|1 －3＝32 3. 4－x 2d x ＝( ) A ．4π B ．2π C ．π [答案] C [解析] 令y ＝4－x 2，则x 2＋y 2＝4(y ≥0)，由定积分的几何意义知所求积分为图中阴影部分的面积， ∴S ＝1 4×π×22＝π. 4.已知甲、乙两车由同一起点同时出发，并沿同一路线(假定为直线)行驶．甲车、乙车的速度曲线分别为v 甲和v 乙(如图所示)．那么对于图中给定的t 0和t 1，下列判断中一定正确的是( ) A ．在t 1时刻，甲车在乙车前面 B ．在t 1时刻，甲车在乙车后面 C ．在t 0时刻，两车的位置相同 D ．t 0时刻后，乙车在甲车前面 [答案] A [解析] 判断甲、乙两车谁在前，谁在后的问题，实际上是判断在t 0，t 1时刻，甲、乙两车行驶路程的大小问题．根据定积分的几何意义知：车在某段时间内行驶的路程就是该时间段内速度函数的定积

微积分基本定理的证明

理学院 School of Sciences 微积分基本定理的证明 Proof of the fundamental theorem of calculus 学生姓名：张智学生学号：201001164 所在班级：数学101 所在专业：数学与应用数学指导老师：杨志林

摘要微积分学这门学科在数学发展中的地位是十分重要的，自十七世纪以来，微积分不断完善成为一门学科。而微积分基本定理的则是微积分中最重要的定理，它的建立标志着微积分的完成，成为数学发展史的一个里程碑。因此就有了研究微积分基本定理的必要性。本文从十七世纪到二十世纪以来的科学家如巴罗、牛顿、莱布尼兹、柯西、黎曼、勒贝格等人对微积分基本定理的发展所作出的贡献展开论述。并论述了定理在微积分学理论发展中的应用。如换元公式、分部积分公式、Taylor中值定理的积分证明、连续函数的零点定理的证明，建立了微分中值定理与积分中值定理的联系，在一元函数和多元函数上的推广等等。最后给出定理的几个证明方法。关键词：微积分基本定理，发展史，定理的应用，定理的证明

ABSTRACT Calculus the subject in the position of the development of mathematics is very important,since seventeenth Century,calculus constantly improved as a discipline.While the fundamental theorem of calculus is the most important theorems in calculus,which establishment marks the complete of the calculus, become a milepost of the development history of mathematics. So it is necessary to study the fundamental theorem of calculus. In this paper,since seventeenth Century to twentieth Century,launches the elaboration from scientists such as Barrow, Newton, Leibniz, Cauchy, Riemann, Lebesgue and others on made the contribution to the development of the fundamental theorem of calculus. And discusses the application of theorem in the development of the calculus theory.Such as the transform formula, integral formula of integration by parts, proof of the Taylor mean value theorem of continuous function, the zero point theorem proof, established the differential mean value theorem and the integral mean value theorem in contact,a unary function and multivariate function on the promotion and so on.Finally gave several proofs of the theorem. Keywords:Fundamental Theorem of Calculus,phylogeny,Application,Proof

7.微积分基本定理练习题

7、微积分基本定理一、选择题 1.??0 1(x 2 ＋2x )d x 等于( ) A.13 B.23 C ．1 D.43 2．∫2π π(sin x －cos x )d x 等于( ) A ．－3 B ．－2 C ．－1 D ．0 3．自由落体的速率v ＝gt ，则落体从t ＝0到t ＝t 0所走的路程为( ) A.13gt 20 B ．gt 2 0 C.12gt 20 D.16gt 20 4．曲线y ＝cos x ? ????0≤x ≤3π2与坐标轴所围图形的面积是( ) A ．4 B ．2 C.5 2 D ．3 5．如图，阴影部分的面积是( ) A ．2 3 B ．2－ 3 C.323 D.35 3 6.??0 3|x 2－4|d x ＝( ) A.213 B.223 C.233 D.25 3 7．??241 x d x 等于( ) A ．－2ln2 B ．2ln2 C ．－ln2 D ．ln2 8．若??1a ? ?? ??2x ＋1x d x ＝3＋ln2，则a 等于( ) A ．6 B ．4 C ．3 D ．2 9．(2010·山东理，7)由曲线y ＝x 2 ，y ＝x 3 围成的封闭图形面积为( ) A.112 B.14 C.13 D.7 12 10．设f (x )＝??? ?? x 2 0≤x <12－x 1

11．从如图所示的长方形区域内任取一个点M (x ，y )，则点M 取自阴影部分的概率为________． 12．一物体沿直线以v ＝1＋t m/s 的速度运动，该物体运动开始后10s 内所经过的路程是________． 13．求曲线y ＝sin x 与直线x ＝－π2，x ＝5 4π，y ＝0所围图形的面积为________． 14．若a ＝??02x 2 d x ，b ＝??02x 3 d x ，c ＝??0 2sin x d x ，则a 、b 、c 大小关系是________．三、解答题 15．求下列定积分： ①??0 2(3x 2＋4x 3 )d x ； ② sin 2 x 2 d x . 17．求直线y ＝2x ＋3与抛物线y ＝x 2 所围成的图形的面积． 18．(1)已知f (a )＝??0 1(2ax 2 －a 2 x )d x ，求f (a )的最大值； (2)已知f (x )＝ax 2 ＋bx ＋c (a ≠0)，且f (－1)＝2，f ′(0)＝0，??0 1f (x )d x ＝－2，求a ，b ，c 的值． DBCDCCDDAC 11. 13 12. 23(1132－1) 13.4－2 2 [解析] 所求面积为＝1＋2＋? ?? ?? 1－22＝4－22. 14.[答案] c