diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 3cfdbbc..d0280ce 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -1003,7 +1003,7 @@ "_journal/2024-12/2024-12-04.md": "965f6619edf1002d960203e3e12a413b", "_journal/2024-12-06.md": "d75323d0fec57f4fc1f13cb4370df18d", "_journal/2024-12/2024-12-05.md": "4f3b1e7a43e01cc97b0eed6fbc6c1f96", - "calculus/integrals.md": "1d34fb199b962f61cdc22f350817d5c3", + "calculus/integrals.md": "7c10ec02401c982039ed421c4435c0ad", "_journal/2024-12-07.md": "bfb6c4db0acbacba19f03a04ec29fa5c", "_journal/2024-12/2024-12-06.md": "d73b611d2d15827186a0252d9b9a6580", "_journal/2024-12-08.md": "5662897539b222db1af45dcd217f0796", @@ -1077,7 +1077,9 @@ "_journal/2024-12-29.md": "e7808872f56a12b51165fc86a1c48e60", "_journal/2024-12/2024-12-28.md": "1ad3caec4ea6f597cc5156f19b274c50", "data-models/rdf/sparql.md": "579cede269025cde2314c3052f272367", - "data-models/rdf/index.md": "979fa61c449648774438c4f29f782602" + "data-models/rdf/index.md": "979fa61c449648774438c4f29f782602", + "_journal/2024-12-30.md": "69762de7ad0fe8ee02b9daaaa2c1745c", + "_journal/2024-12/2024-12-29.md": "e7808872f56a12b51165fc86a1c48e60" }, "fields_dict": { "Basic": [ diff --git a/notes/calculus/integrals.md b/notes/calculus/integrals.md index c895d4b..79ee4df 100644 --- a/notes/calculus/integrals.md +++ b/notes/calculus/integrals.md @@ -11,16 +11,128 @@ tags: The integral is usually defined first in terms of step functions and then general ordinate sets. It is closely tied to [[area]]. In particular, the integral of some nonnegative function on a closed interval is defined so that its area is equal to the area of the ordinate set in question. +Suppose $f$ is [[#Integrable Functions|integrable]] on interval $[a, b]$. Then the **integral** of $f$ from $a$ to $b$ is denoted as $$\int_a^b f(x) \,dx.$$ + +The **lower limit of integration** is $a$. The **upper limit of integration** is $b$. Together they form the **limits of integration**. $f(x)$ is called the **integrand** whereas $dx$ is called the **differential**. Furthermore, we define $$\int_a^b f(x) \,dx = -\int_b^a f(x) \,dx \quad\text{and}\quad \int_a^a f(x)\,dx = 0.$$ + +%%ANKI +Basic +How is the integral of $f$ from $a$ to $b$ denoted? +Back: $\int_a^b f(x) \,dx$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What is $\int_a^b f(x) \,dx$ called? +Back: The integral of $f$ from $a$ to $b$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Integral $\int_a^b f(x) \,dx$ is assumed to be defined on what interval? +Back: Closed interval $[a, b]$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Let $f$ be integrable over $[a, b]$. How is $\int_b^a f(x) \,dx$ defined? +Back: As $-\int_a^b f(x) \,dx$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Let $f$ be integrable over $[a, b]$. What does $\int_a^b f(x) \,dx$ evaluate to after swapping limits of integration? +Back: $-\int_b^a f(x) \,dx$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Let $f$ be integrable over $[a, b]$. What does $\int_a^a f(x) \,dx$ evaluate to? +Back: $0$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What name is given to $a$ in $\int_a^b f(x) \,dx$? +Back: The lower limit of integration. +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + +%%ANKI +Basic +What does the lower limit of integration refer to in $\int_a^b f(x) \,dx$? +Back: $a$ +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + +%%ANKI +Basic +What name is given to $b$ in $\int_a^b f(x) \,dx$? +Back: The upper limit of integration. +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + +%%ANKI +Basic +What name is given collectively to $a$ and $b$ in $\int_a^b f(x) \,dx$? +Back: The limits of integration. +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + +%%ANKI +Basic +What name is given to $f(x)$ in $\int_a^b f(x) \,dx$? +Back: The integrand. +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + +%%ANKI +Basic +What does the integrand refer to in $\int_a^b f(x) \,dx$? +Back: $f(x)$ +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + +%%ANKI +Basic +What name is given to $dx$ in $\int_a^b f(x) \,dx$? +Back: The differential. +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + +%%ANKI +Basic +What does the differential refer to in $\int_a^b f(x) \,dx$? +Back: $dx$ +Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). + +END%% + ## Step Functions Let $s$ be a step function defined on [[intervals|interval]] $[a, b]$, and let $P = \{x_0, x_1, \ldots, x_n\}$ be a [[intervals#Partitions|partition]] of $[a, b]$ such that $s$ is constant on the open subintervals of $P$. Denote by $s_k$ the constant value that $s$ takes in the $k$th open subinterval, so that $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k, \quad k = 1, 2, \ldots, n.$$ The **integral of $s$ from $a$ to $b$**, denoted by the symbol $\int_a^b s(x)\,dx$, is defined by the following formula: $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$ -Furthermore, $$\int_a^b s(x) \,dx = -\int_b^a s(x) \,dx$$ - -and $$\int_a^a s(x)\,dx = 0.$$ - %%ANKI Basic Apostol first introduces the integral for the ordinate sets of what kind of function? @@ -31,36 +143,12 @@ END%% %%ANKI Basic -How is the integral of $s$ from $a$ to $b$ denoted? -Back: $\int_a^b s(x) \,dx$ -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - -%%ANKI -Basic -Let $s$ be a step function. How is the integral of $s$ from $a$ to $b$ defined? -Back: Given partition $P = \{x_0, x_1, \ldots, x_n\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$ +Let $s$ be a step function. How is $\int_a^b s(x) \,dx$ defined? +Back: Given partition $P = \{x_0 = a, x_1, \ldots, x_n = b\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% -%%ANKI -Basic -What is $\int_a^b s(x) \,dx$ called? -Back: The integral of $s$ from $a$ to $b$. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - -%%ANKI -Basic -Integral $\int_a^b s(x) \,dx$ is assumed to be defined on what interval? -Back: Closed interval $[a, b]$. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - %%ANKI Basic Let $s$ be a step function. $\int_a^b s(x) \,dx$ corresponds to what big operator? @@ -119,7 +207,7 @@ END%% %%ANKI Basic -Let $s$ be a step function. How does $\int_a^b s(x) \,dx$ relate to refinements of $s$'s partition? +Let $s$ be a step function. How does $\int_a^b s(x) \,dx$ change as $s$'s partition is refined? Back: N/A. Its value does not change. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). @@ -127,8 +215,8 @@ END%% %%ANKI Basic -Let $s$ be a constant function. What does $\int_a^b s(x) \,dx$ evaluate to? -Back: $c(b - a)$ where $s(x) = c$ for all $x \in [a, b]$. +Let $f$ be a constant function. What does $\int_a^b f(x) \,dx$ evaluate to? +Back: $c(b - a)$ where $f(x) = c$ for all $x \in [a, b]$. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% @@ -140,106 +228,62 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int END%% -%%ANKI -Basic -Let $s$ be a step function over $[a, b]$. How is $\int_b^a s(x) \,dx$ defined? -Back: As $-\int_a^b s(x) \,dx$. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - %%ANKI Basic Let $s$ be a step function over $[a, b]$. How is $\int_a^b s(x) \,dx$ defined? -Back: Given partition $P = \{x_0, x_1, \ldots, x_n\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$ +Back: Given partition $P = \{x_0 = a, x_1, \ldots, x_n = b\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% -%%ANKI -Basic -Let $s$ be a step function over $[a, b]$. What does $\int_a^b s(x) \,dx$ evaluate to after swapping limits of integration? -Back: $-\int_b^a s(x) \,dx$. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% +## Integrable Functions + +TODO + +### Integrand Additivity + +Let $f$ and $g$ be integrable over $[a, b]$. Then $$\int_a^b f(x) + g(x) \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx$$ %%ANKI Basic -Let $s$ be a step function over $[a, b]$. What does $\int_a^a s(x) \,dx$ evaluate to? -Back: $0$. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - -%%ANKI -Basic -Let $s$ be a step function over $[a, b]$. What name is given to $a$ in $\int_a^b s(x) \,dx$? -Back: The lower limit of integration. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - -%%ANKI -Basic -Let $s$ be a step function over $[a, b]$. What name is given to $b$ in $\int_a^b s(x) \,dx$? -Back: The upper limit of integration. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - -%%ANKI -Basic -Let $s$ be a step function over $[a, b]$. What name is given to $a$ and $b$ in $\int_a^b s(x) \,dx$? -Back: The limits of integration. -Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). - -END%% - -### Additivity - -Let $s$ and $t$ be step functions defined on $[a, b]$. Then $$\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$$ - -%%ANKI -Basic -Let $s$ and $t$ be step functions over $[a, b]$. What does the additive property of integrals state? -Back: $\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$ +What does the additivity property w.r.t. the integrand state? +Back: Let $f$ and $g$ be integrable over $[a, b]$. Then $\int_a^b f(x) + g(x) \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx$. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% %%ANKI Basic -Let $s$ and $t$ be step functions over $[a, b]$. What is the following identity called? $$\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$$ +What is the following identity called? $$\int_a^b f(x) + g(x) \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx$$ -Back: The additive property. +Back: The additive property w.r.t. the integrand. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% %%ANKI Basic -Let $s$ and $t$ be step functions over $[a, b]$. How is the following more compactly written? $$\int_a^b s(x) \,dx + \int_a^b t(x) \,dx$$ -Back: $\int_a^b s(x) + t(x) \,dx$ +Let $f$ and $g$ be integrable over $[a, b]$. How is the following more compactly written? $$\int_a^b f(x) \,dx + \int_a^b g(x) \,dx$$ +Back: As $\int_a^b f(x) + g(x) \,dx$. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% ### Homogeneousness -Let $s$ be a step function defined on $[a, b]$. Let $c \in \mathbb{R}$. Then $$\int_a^b c \cdot s(x) \,dx = c\int_a^b s(x) \,dx$$ +Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $$\int_a^b c \cdot f(x) \,dx = c\int_a^b f(x) \,dx$$ %%ANKI Basic -Let $s$ be a step function over $[a, b]$. What does the homogeneous property of integrals state? -Back: For all $c \in \mathbb{R}$, $\int_a^b c \cdot s(x) \,dx = c \int_a^b s(x) \,dx$. +What does the homogeneous property of integrals state? +Back: Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $\int_a^b c \cdot f(x) \,dx = c \int_a^b f(x) \,dx$. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% %%ANKI Basic -Let $s$ be a step function defined over $[a, b]$ and $c \in \mathbb{R}$. What is the following identity called? $$\int_a^b c \cdot s(x) \,dx = c\int_a^b s(x) \,dx$$ +What is the following identity called? $$\int_a^b c \cdot f(x) \,dx = c\int_a^b f(x) \,dx$$ Back: The homogeneous property. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). @@ -248,19 +292,19 @@ END%% ### Linearity -Let $s$ and $t$ be step functions defined on $[a, b]$. Let $c_1, c_2 \in \mathbb{R}$. Then $$\int_a^b [c_1s(x) + c_2t(x)] \,dx = c_1 \int_a^b s(x) \,dx + c_2 \int_a^b t(x) \,dx$$ +Let $f$ and $g$ be integrable over $[a, b]$. Let $c_1, c_2 \in \mathbb{R}$. Then $$\int_a^b [c_1f(x) + c_2g(x)] \,dx = c_1 \int_a^b f(x) \,dx + c_2 \int_a^b g(x) \,dx$$ %%ANKI Basic -Let $s$ and $t$ be step functions over $[a, b]$ and $c_1, c_2 \in \mathbb{R}$. What does the linearity property of integrals state? -Back: $\int_a^b [c_1 s(x) + c_2 t(x)] \,dx = c_1 \int_a^b s(x) \,dx + c_2 \int_a^b t(x) \,dx$ +What does the linearity property of integrals state? +Back: Let $f$ and $g$ be integrable over $[a, b]$ and $c_1, c_2 \in \mathbb{R}$. Then $$\int_a^b [c_1 f(x) + c_2 g(x)] \,dx = c_1 \int_a^b f(x) \,dx + c_2 \int_a^b g(x) \,dx$$ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% %%ANKI Basic -Let $s$ and $t$ be step functions over $[a, b]$ and $c_1, c_2 \in \mathbb{R}$. What is the following identity called? $$\int_a^b [c_1s(x) + c_2t(x)] \,dx = c_1 \int_a^b s(x) \,dx + c_2 \int_a^b t(x) \,dx$$ +What is the following identity called? $$\int_a^b [c_1f(x) + c_2g(x)] \,dx = c_1 \int_a^b f(x) \,dx + c_2 \int_a^b g(x) \,dx$$ Back: The linearity property. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). @@ -269,33 +313,33 @@ END%% %%ANKI Basic The linearity property is immediately derived from what other two properties? -Back: The additive and homogeneous properties. +Back: The additive property w.r.t. the integrand and homogeneousness. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% %%ANKI Cloze -The {linearity} property of integrals is a combination of the {additive} and {homogenous} properties. +The {linearity} property of integrals is an immediate consequence of {additivity w.r.t the integrand} and {homogenousness}. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% ### Comparison Theorem -Let $s$ and $t$ be step functions defined on $[a, b]$. Suppose $s(x) < t(x)$ for all $x \in [a, b]$. Then $$\int_a^b s(x) \,dx < \int_a^b t(x) \,dx$$ +Let $f$ and $b$ be integrable over $[a, b]$. If $f(x) \leq g(x)$ for all $x \in [a, b]$, then $$\int_a^b f(x) \,dx \leq \int_a^b g(x) \,dx$$ %%ANKI Basic -Let $s$ and $t$ be step functions over $[a, b]$. What does the comparison theorem of integrals state? -Back: If $s(x) < t(x)$ for all $x \in [a, b]$, then $\int_a^b s(x) \,dx < \int_a^b t(x) \,dx$. +What does the comparison theorem for integrals state? +Back: Let $f$ and $g$ be integrable over $[a, b]$. If $f(x) \leq g(x)$ for all $x \in [a, b]$, then $$\int_a^b f(x) \,dx \leq \int_a^b g(x) \,dx.$$ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% %%ANKI Basic -Let $s$ and $t$ be step functions over $[a, b]$ such that $s(x) < t(x)$ for all $x \in [a, b]$. What is the following called? $$\int_a^b s(x) \,dx < \int_a^b t(x) \,dx$$ +Let $f$ and $g$ be integrable over $[a, b]$ such that $f(x) \leq g(x)$ for all $x \in [a, b]$. What is the following called? $$\int_a^b f(x) \,dx \leq \int_a^b g(x) \,dx$$ Back: The comparison theorem. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). @@ -303,20 +347,66 @@ END%% %%ANKI Basic -The comparison theorem of step function integrals corresponds to what property of area? -Back: The monotone property of area. +The comparison theorem of integrals corresponds to what property of area? +Back: The monotone property. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% %%ANKI Basic -The monotone property of area corresponds to what theorem of step function integrals? +The monotone property of area corresponds to what basic property of integrals? Back: The comparison theorem. Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). END%% +### Interval of Integration Additivity + +If two of the following three integrals exist, the third also exists, and we have $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx = \int_a^c f(x) \,dx$$ + +%%ANKI +Basic +What does the additivity property w.r.t. the interval of integration state? +Back: If two of the following three integrals exist, the third also exists, and we have $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx = \int_a^c f(x) \,dx$$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Let $f$ be integrable over an interval containing $a$, $b$, and $c$. What is the following identity called? $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx = \int_a^c f(x) \,dx$$ + +Back: The additive property w.r.t. the interval of integration. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Assume the following integrals exist. How is the following written more compactly? $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx$$ +Back: $\int_a^c f(x) \,dx$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +The additvity theorem w.r.t. intervals of integration corresponds to what property of area? +Back: The additive property of area. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +The additive property of area corresponds to what basic property of integrals? +Back: The additive property w.r.t. the interval of integration. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + ## Bibliography +* “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875). * Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). \ No newline at end of file