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CH 5

CH 5. Integration 積分. 學習內容. 面積與距離. Area and Distance The Definite Integral The Fundamental Theorem of Calculus Indefinite Integrals and the Net Change Theorem Tne Substitution Rule. 定積分. 微積分基本定理. 不定積分. 代換法. 5.2. The Definite Integral 定積分. 學習重點. 用定積分符號表示不規則區域的面積 知道均值定理

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CH 5

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  1. CH 5 Integration 積分

  2. 學習內容 面積與距離 • Area and Distance • The Definite Integral • The Fundamental Theorem of Calculus • Indefinite Integrals and the Net Change Theorem • Tne Substitution Rule 定積分 微積分基本定理 不定積分 代換法

  3. 5.2 The Definite Integral 定積分

  4. 學習重點 • 用定積分符號表示不規則區域的面積 • 知道均值定理 • 利用均值定理求函數的平均值 • 知道廣義的定積分定義

  5. 面積與定積分的關係

  6. Example 1 Expressas an integral on the interval [0, π].

  7. 什麼情形下有定積分 ? 若函數在[a, b]是連續的， 則函數在[a, b]的定積分存在。 連續函數在[a, b]是可積分的。

  8. 魔術手法－如何將不規則面積變規則? = 面積 = 22π/2 = 2π 面積 = 4．π/2 = 2π 平均值

  9. 積分均值定理 平均值 ．面積 1/寬

  10. X軸下方的面積 上(正)下(負)面積抵銷! f(ck) > 0 面積 > 0 f(ck) < 0 面積 < 0

  11. Example 4 • Evaluate the following integrals by interpreting each in terms of areas. • a. • b. = ? = ?

  12. y2 = 1 - x2 x2 + y2 = 1 單位圓

  13. y = x– 1 Figure 5.2.10, p. 306

  14. 積分的性質 零面積 係數法則 加法法則 優勢性 加法性 不等性

  15. Example 7 • If it is known that • Find = ?

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