5.1 Areas Between Curves

1. 定義：面積

     連續函數\(\,f\,\)圖形下的區域\(\,S\,\)，其面積為\(\,A\,\)

\[A=\lim_{n \to \infty} \sum_{i=1}^{n} \left[f(x_i^*)-g(x_i^*)\right]\Delta x \]

     OpenStax https://openstax.org/books/calculus-volume-1/pages/6-1-areas-between-curves


2. 定理：面積 (對 x 積分)

     若 \(f\) 及 \(g\) 在 \([a, b]\) 上連續，且 \(f(x) ≥ g(x) \;\forall\; x \in [a,b]\)，則曲線 \(y = f(x)\) 和 \(y = g(x)\)，及直線 \(x=a\) 和 \(x=b\)
     所圍的區域面積 \(A\) 為

\[A=\int_{a}^{b}\left[f(x)-g(x)\right]\;dx\]

3. 定理：面積

     曲線 \(y = f(x)\) 和 \(y = g(x)\)，及直線 \(x=a\) 和 \(x=b\) 所圍的區域面積 \(A\) 為

\[A=\int_{a}^{b}|f(x)-g(x)|\;dx\]

4. 例題 (GeoGebra)

     (1) 例題1：https://www.geogebra.org/m/c5JyQnxE
     (2) 例題3：https://www.geogebra.org/m/hk7NSNx7


5. 定理：面積 (對 y 積分)

     若 \(f\) 及 \(g\) 在 \([c, d]\) 上連續，且 \(f(y) ≥ g(y) \;\forall\; y \in [c,d]\)，則曲線 \(x = f(y)\) 和 \(x = g(y)\)，及直線 \(y=c\) 和 \(y=d\)
     所圍的區域面積 \(A\) 為

\[A=\int_{c}^{d}\left[f(y)-g(y)\right]\;dy\]

     Paul’s Online Notes https://tutorial.math.lamar.edu/classes/calci/areabetweencurves.aspx
     The LibreTexts libraries Mathematics
     https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/06%3A_Applications_of_Integration/6.01%3A_Areas_between_Curves


5.2 Volumes

1. 定義：體積

     令 \(S\) 為介於 \(x = a\) 及 \(x = b\) 之間的立體區域。 \(P_x\) 是通過 \(x\)-座標為 \(x\), 且與\(x\)-軸垂直之平面。
     假設 \(S\) 在 \(P_x\) 的截面積是 \(A(x)\)，\(A(x)\) 為連續函數，則 \(S\) 的體積為 \(V\)

\[V=\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\Delta x=\int_{a}^{b}A(x)\;dx\]

     OpenStax https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing


3. 例題 (GeoGebra)

     (1) 例題2：https://www.geogebra.org/m/a8rfdptg
     (2) 例題4：https://www.geogebra.org/m/Hw7bnB2q


4. 截面 (Cross Section) 求體積

     (1) 例題7和例題9：https://www.geogebra.org/m/heST6SaC
     (2) 例題8：https://www.geogebra.org/m/xDNjSjEK#material/JW4RFRbs


     Math24 https://math24.net/volume-solid-with-known-cross-section.html


5.3 Volumes by Cylindrical Shells

1. 圓柱殼 (Cylindrical Shells)

\[\lim_{n \to \infty} \sum_{i=1}^{n} \;2\pi\,\overline{x}_i\,f(\overline{x}_i)\Delta x=\int_{a}^{b}2\pi x\,f(x)\;dx\]

     The LibreTexts libraries Mathematics
     https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/06%3A_Applications_of_Integration/6.03%3A_Volumes_of_Revolution_-_Cylindrical_Shells
      OpenStax https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells
      Simon Fraser University https://www.sfu.ca/math-coursenotes/Math%20158%20Course%20Notes/sec_shell.html


2. 定理

     令 \(A\) 為 \(y = f(x) ≥ 0\) ， \(a ≤ x ≤ b\) 與 \(x\)-軸及 \(x = a\)， \(x = b\) 所圍成的區域。將 A 繞 \(y\)-軸旋轉，
     則旋轉體體積為

\[V=\int_{a}^{b}2\pi xf(x)\;dx \quad 0≤a<b\]

3. 例題 (GeoGebra)

     (1) 例題1：https://www.geogebra.org/m/ggfye7dj
     (2) 例題3：https://www.geogebra.org/m/dEw6qQKX


5.5 Average Value of a Function

1. 積分均值定理 (The Mean Value Theorem for Integrals)

     若 \(f\) 在 \([a, b]\) 上連續，\(\exists \;c \in [a, b]\) 使得

\[f(c)=f_{avg}=\frac {1}{b-a}\int_{a}^{b}f(x)\;dx\]

     也就是

\[\int_{a}^{b}f(x)\;dx=f(c)(b-a)\]

     Paul’s Online Notes https://tutorial.math.lamar.edu/classes/calci/avgfcnvalue.aspx
     The LibreTexts libraries Mathematics
     https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Seeburger)/05%3A_Integration/5.03%3A_The_Fundamental_Theorem_of_Calculus



Reference :

Essential Calculus, metric edition 2e, (2022) James Stewart, Daniel K. Clegg, Saleem Watson, Cengage Learning.
Calculus Early Transcendentals:Differential & Multi-Variable Calculus for Social Sciences (2017). Petra Menz, Nicola Mulberry from Lyryx’ textbook Paul’s Online Notes
The LibreTexts libraries Methmatics
Math’s Fun Advance
Math24
OpenStax
GeoGebra