Reorganize integral property flashcards.

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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).
+<!--ID: 1733520215063-->
+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).
+<!--ID: 1733520215070-->
+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).
+<!--ID: 1733520215074-->
+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).
+<!--ID: 1734816555507-->
+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).
+<!--ID: 1734816555515-->
+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).
+<!--ID: 1734816555518-->
+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).
+<!--ID: 1734816555521-->
+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).
+<!--ID: 1735613867214-->
+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).
+<!--ID: 1734816555523-->
+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).
+<!--ID: 1734816555526-->
+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).
+<!--ID: 1735613867216-->
+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).
+<!--ID: 1735613867218-->
+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).
+<!--ID: 1735613867219-->
+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).
+<!--ID: 1735613867220-->
+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?
+Let $s$ be a step function. How is $\int_a^b s(x) \,dx$ defined?
-Back: $\int_a^b s(x) \,dx$
+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).
-<!--ID: 1733520215063-->
-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, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1733520215067-->
 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).
-<!--ID: 1733520215070-->
-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).
-<!--ID: 1733520215074-->
-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).
 <!--ID: 1733520215118-->
@@ -127,8 +215,8 @@ END%%
 
 %%ANKI
 Basic
-Let $s$ be a constant function. What does $\int_a^b s(x) \,dx$ evaluate to?
+Let $f$ be a constant function. What does $\int_a^b f(x) \,dx$ evaluate to?
-Back: $c(b - a)$ where $s(x) = c$ for all $x \in [a, b]$.
+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).
 <!--ID: 1733520215126-->
 END%%
@@ -140,106 +228,62 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int
 <!--ID: 1733520215132-->
 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).
-<!--ID: 1734816555507-->
-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).
 <!--ID: 1734816555512-->
 END%%
 
-%%ANKI
+## Integrable Functions
-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?
+TODO
-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).
+### Integrand Additivity
-<!--ID: 1734816555515-->
+
-END%%
+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?
+What does the additivity property w.r.t. the integrand state?
-Back: $0$.
+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).
-<!--ID: 1734816555518-->
-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).
-<!--ID: 1734816555521-->
-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).
-<!--ID: 1734816555523-->
-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).
-<!--ID: 1734816555526-->
-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: $\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1734814463659-->
 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).
 <!--ID: 1734814463668-->
 END%%
 
 %%ANKI
 Basic
-Let $s$ and $t$ be step functions over $[a, b]$. How is the following more compactly written? $$\int_a^b s(x) \,dx + \int_a^b t(x) \,dx$$
+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: $\int_a^b s(x) + t(x) \,dx$
+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).
 <!--ID: 1734814463673-->
 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?
+What does the homogeneous property of integrals state?
-Back: For all $c \in \mathbb{R}$, $\int_a^b c \cdot s(x) \,dx = c \int_a^b s(x) \,dx$.
+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).
 <!--ID: 1734814463679-->
 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?
+What does the linearity property of integrals state?
-Back: $\int_a^b [c_1 s(x) + c_2 t(x)] \,dx = c_1 \int_a^b s(x) \,dx + c_2 \int_a^b t(x) \,dx$
+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).
 <!--ID: 1734814463699-->
 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).
 <!--ID: 1734814463704-->
@@ -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).
 <!--ID: 1734814463710-->
 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).
 <!--ID: 1734814463693-->
 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?
+What does the comparison theorem for integrals state?
-Back: If $s(x) < t(x)$ for all $x \in [a, b]$, then $\int_a^b s(x) \,dx < \int_a^b t(x) \,dx$.
+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).
 <!--ID: 1734815755275-->
 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).
 <!--ID: 1734815755282-->
@@ -303,20 +347,66 @@ END%%
 
 %%ANKI
 Basic
-The comparison theorem of step function integrals corresponds to what property of area?
+The comparison theorem of integrals corresponds to what property of area?
-Back: The monotone 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).
 <!--ID: 1734815755285-->
 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).
 <!--ID: 1734815755288-->
 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).
+<!--ID: 1735613867221-->
+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).
+<!--ID: 1735613867222-->
+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).
+<!--ID: 1735613867223-->
+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).
+<!--ID: 1735613867224-->
+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).
+<!--ID: 1735613867225-->
+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).