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# Chapter 4 Integration (積分)

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## (積分)

### Fall 2018

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 1/119

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## 本章預定授課範圍

### 4.6 Indeterminate Forms and L’Hôpital’s Rule4.7 The Natural Logarithmic Function: Integration4.8 Inverse Trigonometric Functions: Integration

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 2/119

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## (反導函數與不定積分)

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### The functions F1(x) = x3, F2(x) = x3− 5 and F 3 (x) = x3 + 97 are antiderivatives of f(x) = 3x2 on R.

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### G is an antiderivative of f on I⇐⇒ G(x) = F(x) + C,where C is a constant.

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## Indefinite Integration (不定積分)

### where f is called the integrand (被積函數) and C is called the constant of integration. (積分常數)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 7/119

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### = F′(x) = f(x), i.e., the integration and differentiation are inverse operations to each other! (積分與微分互為反運算!)

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### n + 1xn+1+ C forn̸= −1 .

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### csc x cot x dx =− csc x + C.

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### x−1dx = ln|x| + C.

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### k2x2− 1dx = sec −1(kx)+ C.

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## Application of the Indefinite Integral

### How to solve the particular solution (特解) of the I.V.P. (1)?

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 17/119

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## Solutions of the I.V.P.

### F(x) = Fp(x) = G(x) + y0− G(x 0 ).

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## Solutions of the I.V.P.

### F(x) = Fp(x) = G(x) + y 0 − G(x 0 ).

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 18/119

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## Section 4.2Area(面積)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 21/119

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### where i is the index (足碼) of the summation and aiis the ith term(第 i 項) of the sum.

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### = n2(n+1)42 .

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## Area of a Plane Region

### R = {(x, y) ∈ R2| a ≤ x ≤ b, 0 ≤ y ≤ f(x)}.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 24/119

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## Lower and Upper Sums (1/2)

### f(mi ) ≤ f(x) ≤ f(Mi ) ∀ x ∈ [xi−1, xi].

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## Lower and Upper Sums (1/2)

### f(mi ) ≤ f(x) ≤ f(Mi ) ∀ x ∈ [xi−1, xi].

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 25/119

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## Lower and Upper Sums (2/2)

### From the definitions of s(n) and S(n), we see thats(n)≤ area(R) ≤ S(n) ∀ n ∈ N.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 26/119

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## 示意圖 (承上頁)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 27/119

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### In this case, we know that area( R) = A ∃ by the Squeeze Thm.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 32/119

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### where ci∈ [xi−1, xi] for i = 1, 2, . . . , n and ∆x =b−an .

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## Riemann Sums and Definite Integrals(黎曼和與定積分)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 37/119

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## Riemann Sums (黎曼和)

### f(ci)∆xi is called a Riemann sum (黎曼和) of f for the partition ∆.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 38/119

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### So, ∥∆∥ → 0 ⇒ n → ∞ , but n→ ∞ ; ∥∆∥ → 0 ! (2) If ∆xi = b−an∀ i , then ∥∆∥ → 0 ⇐⇒ n → ∞ .

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 39/119

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## Definite Integrals (定積分)

### (2) The number a is called the lower limit of integration (積分下限) and b is called the upper limit of integration (積分上限).

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### f(w) dw =· · ·denote the same definite integral of f from a to b.

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## Existence of Definite Integrals

### for any partition ∆ of the interval I.

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### (函數 f 在 [a, b] 上最多僅有限個不連續點 =⇒ f 在 [a, b] 上必定 是一個黎曼可積的函數!)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 43/119

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### f(x) dx≥ 0 .

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### f(x) dx.

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## Properties of Definite Integrals

### g(x) dx.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 48/119

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## 示意圖 (承上頁)

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### ag(x) dx.

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## The Fundamental Theorem of Calculus(微積分基本定理; F.T.C.)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 53/119

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### Note: the definite integral of f from a to b can be evaluated bythe function values of antiderivative F(a) and F(b) directly!

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### −2 = 1 2− (−2)2 = 1 − 4 = −3,since F(x) = x2is an antiderivative of the integrand f(x) = 2x.

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### f(x) dx.

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## Proof of Thm 4.10

### f(x) dx≡ fav.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 59/119

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## 示意圖 (承上頁)

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### f(x) dx.

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## The Definite Integral as a Function

### Furthermore, is F an antiderivative of f on I?

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 63/119

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### = f(x)∀ x ∈ I,i.e. F is an antiderivative of f on the open interval I.

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## Proof of Thm 4.11

+

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### f(c 2 ) = f(x),since f is conti. on I. This completes the proof.

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### = −f(x)for all x∈ I. (積分上下限顛倒會差了一個負號喔!)

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### = f(u(x))· u′(x)− f(v(x)) · v′(x).

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## Integration by Substitution(積分代換法)

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### f(u)du, and its associated definite integrals.

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### f(g(x))g′(x) dx = F(g(x)) + C,where C is a constant of integration.

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## Change of Variables for Indefinite Integrals

### f(g(x))g′(x) dx = F(g(x)) + C =F(u) + C,where C is a constant of integration.

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### where C is a constant of integration.

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## Change of Variables for Definite Integrals

### (將對 x 的定積分變數變換成對 u 的定積分!)

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## Integration of Odd and Even Functions

### f(x) dx = 0.

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## 示意圖 (承上頁)

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## 示意圖 (承上例)

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## (不定型與羅必達法則)

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### ∞, 0 · ∞, 1∞,∞0, 00,∞ − ∞, which are called the indeterminate forms (不定型).

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### Note: Thm 4.18 also holds for the one-sided limits. (羅必達法則 也適用於求解單邊極限值!)

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### 1 = cos(0) = 1.

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### ∞).

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### x→ceg(x) ln[f(x)]= eL.

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### x→∞eln xx= e0= 1.

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### (使用通分技巧將原極限問題變成標準不定型!)

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### . (Type ∞ − ∞)

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## Solution of Example 7

### 2 . (直接代入求極限喔!)

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## (自然對數函數的積分)

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### udu = ln|u(x)| + C,where C is a constant of integration.

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### csc u du =− ln | csc u + cot u| + C,where C is a constant of integration.

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## Integration(反三角函數的積分)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 111/119

Thousands of years ago, people in Mexico loved chocolate so much that, at times, they used it as money or for trade.. They even had a god of chocolate

T wo thousand years ago, kings called emperors ruled the city of Rome.. At that time, Rome was one of the strongest cities in

A superhero needs his own special physical abilities to help him fight enemies.. I want to draw my own

Improper integrals can arise on bounded intervals. Suppose that f is continuous on the half-open interval [a, b) but is unbounded there..

Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley &amp; Sons, Inck. All

Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley &amp; Sons, Inc.. All

trigonometric identities will simplify the integrand and make the method of integration obvious... Classify the Integrand According to

requires to be able to evaluate the integrand at arbitrary points making it easy to implement.. arbitrary points, making it easy to implement and applicable to

• Three uniform random numbers are used, the first one determines which BxDF to be sampled and then sample that BxDF using the other two random numbers.. Sampling

• Metropolis sampling can efficiently generate a set of samples from any non negative function f set of samples from any non-negative function f requiring only the ability to

• Three uniform random numbers are used, the first one determines which BxDF to be sampled first one determines which BxDF to be sampled (uniformly sampled) and then sample that

Spectrum Sample_L(Point &amp;p, float pEpsilon, LightSample &amp;ls float time Vector *wi LightSample &amp;ls, float time, Vector *wi, float *pdf, VisibilityTester *vis) = 0;.

4-7 The photocopy of the letter of agreement between Business Incubation Center of Feng Chia University and the applied company over the operations and cultivations.. 4-8

4-7 The photocopy of the letter of agreement between Business Incubation Center of Feng Chia University and the applied company over the operations and cultivations.. 4-8

Some of the most common closed Newton-Cotes formulas with their error terms are listed in the following table... The following theorem summarizes the open Newton-Cotes

6.1 Integration by Parts 6.2 Trigonometric Integrals 6.3 Trigonometric Substitution 6.4 Partial Fractions.. 6.7

We point out that extending the concepts of r-convex and quasi-convex functions to the setting associated with second-order cone, which be- longs to symmetric cones, is not easy

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However, since most of the reports focus more on the assessments of the ecological function and landscape integration, and less on the factors affecting the

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