Reorganize integral property flashcards.
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},
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"fields_dict": {
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"Basic": [
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@ -11,16 +11,128 @@ tags:
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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.
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Suppose $f$ is [[#Integrable Functions|integrable]] on interval $[a, b]$. Then the **integral** of $f$ from $a$ to $b$ is denoted as $$\int_a^b f(x) \,dx.$$
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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 $$\int_a^b f(x) \,dx = -\int_b^a f(x) \,dx \quad\text{and}\quad \int_a^a f(x)\,dx = 0.$$
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%%ANKI
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Basic
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How is the integral of $f$ from $a$ to $b$ denoted?
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Back: $\int_a^b f(x) \,dx$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215063-->
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END%%
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%%ANKI
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Basic
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What is $\int_a^b f(x) \,dx$ called?
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Back: The integral of $f$ from $a$ to $b$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215070-->
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END%%
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%%ANKI
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Basic
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Integral $\int_a^b f(x) \,dx$ is assumed to be defined on what interval?
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Back: Closed interval $[a, b]$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215074-->
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END%%
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%%ANKI
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Basic
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Let $f$ be integrable over $[a, b]$. How is $\int_b^a f(x) \,dx$ defined?
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Back: As $-\int_a^b f(x) \,dx$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555507-->
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END%%
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%%ANKI
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Basic
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Let $f$ be integrable over $[a, b]$. What does $\int_a^b f(x) \,dx$ evaluate to after swapping limits of integration?
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Back: $-\int_b^a f(x) \,dx$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555515-->
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END%%
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%%ANKI
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Basic
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Let $f$ be integrable over $[a, b]$. What does $\int_a^a f(x) \,dx$ evaluate to?
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Back: $0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555518-->
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END%%
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%%ANKI
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Basic
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What name is given to $a$ in $\int_a^b f(x) \,dx$?
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Back: The lower limit of integration.
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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).
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<!--ID: 1734816555521-->
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END%%
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%%ANKI
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Basic
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What does the lower limit of integration refer to in $\int_a^b f(x) \,dx$?
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Back: $a$
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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).
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<!--ID: 1735613867214-->
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END%%
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%%ANKI
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Basic
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What name is given to $b$ in $\int_a^b f(x) \,dx$?
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Back: The upper limit of integration.
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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).
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<!--ID: 1734816555523-->
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END%%
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%%ANKI
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Basic
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What name is given collectively to $a$ and $b$ in $\int_a^b f(x) \,dx$?
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Back: The limits of integration.
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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).
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<!--ID: 1734816555526-->
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END%%
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%%ANKI
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Basic
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What name is given to $f(x)$ in $\int_a^b f(x) \,dx$?
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Back: The integrand.
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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).
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<!--ID: 1735613867216-->
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END%%
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%%ANKI
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Basic
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What does the integrand refer to in $\int_a^b f(x) \,dx$?
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Back: $f(x)$
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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).
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<!--ID: 1735613867218-->
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END%%
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%%ANKI
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Basic
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What name is given to $dx$ in $\int_a^b f(x) \,dx$?
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Back: The differential.
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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).
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<!--ID: 1735613867219-->
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END%%
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%%ANKI
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Basic
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What does the differential refer to in $\int_a^b f(x) \,dx$?
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Back: $dx$
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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).
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<!--ID: 1735613867220-->
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END%%
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## Step Functions
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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 $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k, \quad k = 1, 2, \ldots, n.$$
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The **integral of $s$ from $a$ to $b$**, denoted by the symbol $\int_a^b s(x)\,dx$, is defined by the following formula: $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$
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Furthermore, $$\int_a^b s(x) \,dx = -\int_b^a s(x) \,dx$$
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and $$\int_a^a s(x)\,dx = 0.$$
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%%ANKI
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Basic
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Apostol first introduces the integral for the ordinate sets of what kind of function?
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@ -31,36 +143,12 @@ END%%
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%%ANKI
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Basic
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How is the integral of $s$ from $a$ to $b$ denoted?
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Back: $\int_a^b s(x) \,dx$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215063-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function. How is the integral of $s$ from $a$ to $b$ defined?
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Back: Given partition $P = \{x_0, x_1, \ldots, x_n\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$
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Let $s$ be a step function. How is $\int_a^b s(x) \,dx$ defined?
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Back: Given partition $P = \{x_0 = a, x_1, \ldots, x_n = b\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215067-->
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END%%
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%%ANKI
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Basic
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What is $\int_a^b s(x) \,dx$ called?
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Back: The integral of $s$ from $a$ to $b$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215070-->
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END%%
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%%ANKI
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Basic
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Integral $\int_a^b s(x) \,dx$ is assumed to be defined on what interval?
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Back: Closed interval $[a, b]$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215074-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function. $\int_a^b s(x) \,dx$ corresponds to what big operator?
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@ -119,7 +207,7 @@ END%%
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%%ANKI
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Basic
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Let $s$ be a step function. How does $\int_a^b s(x) \,dx$ relate to refinements of $s$'s partition?
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Let $s$ be a step function. How does $\int_a^b s(x) \,dx$ change as $s$'s partition is refined?
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Back: N/A. Its value does not change.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215118-->
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@ -127,8 +215,8 @@ END%%
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%%ANKI
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Basic
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Let $s$ be a constant function. What does $\int_a^b s(x) \,dx$ evaluate to?
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Back: $c(b - a)$ where $s(x) = c$ for all $x \in [a, b]$.
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Let $f$ be a constant function. What does $\int_a^b f(x) \,dx$ evaluate to?
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Back: $c(b - a)$ where $f(x) = c$ for all $x \in [a, b]$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1733520215126-->
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END%%
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@ -140,106 +228,62 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int
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<!--ID: 1733520215132-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. How is $\int_b^a s(x) \,dx$ defined?
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Back: As $-\int_a^b s(x) \,dx$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555507-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. How is $\int_a^b s(x) \,dx$ defined?
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Back: Given partition $P = \{x_0, x_1, \ldots, x_n\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$
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Back: Given partition $P = \{x_0 = a, x_1, \ldots, x_n = b\}$ with constant value $s_k$ on the $k$th open subinterval, $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555512-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. What does $\int_a^b s(x) \,dx$ evaluate to after swapping limits of integration?
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Back: $-\int_b^a s(x) \,dx$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555515-->
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END%%
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## Integrable Functions
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TODO
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### Integrand Additivity
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Let $f$ and $g$ be integrable over $[a, b]$. Then $$\int_a^b f(x) + g(x) \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx$$
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. What does $\int_a^a s(x) \,dx$ evaluate to?
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Back: $0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555518-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. What name is given to $a$ in $\int_a^b s(x) \,dx$?
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Back: The lower limit of integration.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555521-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. What name is given to $b$ in $\int_a^b s(x) \,dx$?
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Back: The upper limit of integration.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555523-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. What name is given to $a$ and $b$ in $\int_a^b s(x) \,dx$?
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Back: The limits of integration.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734816555526-->
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END%%
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### Additivity
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Let $s$ and $t$ be step functions defined on $[a, b]$. Then $$\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$$
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%%ANKI
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Basic
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Let $s$ and $t$ be step functions over $[a, b]$. What does the additive property of integrals state?
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Back: $\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$
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What does the additivity property w.r.t. the integrand state?
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Back: Let $f$ and $g$ be integrable over $[a, b]$. Then $\int_a^b f(x) + g(x) \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734814463659-->
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END%%
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%%ANKI
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Basic
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Let $s$ and $t$ be step functions over $[a, b]$. What is the following identity called? $$\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$$
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What is the following identity called? $$\int_a^b f(x) + g(x) \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx$$
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Back: The additive property.
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Back: The additive property w.r.t. the integrand.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734814463668-->
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END%%
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%%ANKI
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Basic
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Let $s$ and $t$ be step functions over $[a, b]$. How is the following more compactly written? $$\int_a^b s(x) \,dx + \int_a^b t(x) \,dx$$
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Back: $\int_a^b s(x) + t(x) \,dx$
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Let $f$ and $g$ be integrable over $[a, b]$. How is the following more compactly written? $$\int_a^b f(x) \,dx + \int_a^b g(x) \,dx$$
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Back: As $\int_a^b f(x) + g(x) \,dx$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1734814463673-->
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END%%
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### Homogeneousness
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Let $s$ be a step function defined on $[a, b]$. Let $c \in \mathbb{R}$. Then $$\int_a^b c \cdot s(x) \,dx = c\int_a^b s(x) \,dx$$
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Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $$\int_a^b c \cdot f(x) \,dx = c\int_a^b f(x) \,dx$$
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%%ANKI
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Basic
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Let $s$ be a step function over $[a, b]$. What does the homogeneous property of integrals state?
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Back: For all $c \in \mathbb{R}$, $\int_a^b c \cdot s(x) \,dx = c \int_a^b s(x) \,dx$.
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What does the homogeneous property of integrals state?
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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?
|
||||
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).
|
||||
<!--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?
|
||||
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).
|
||||
<!--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?
|
||||
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).
|
||||
<!--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).
|
||||
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Reference in New Issue