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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Integrals | Obsidian::STEM | calculus::integral |
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Overview
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 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.
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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).
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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).
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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).
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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).
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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).
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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).
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Step Functions
Let s be a step function defined on intervals [a, b], and let P = \{x_0, x_1, \ldots, x_n\} be a intervals#Partitions 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})
%%ANKI Basic Apostol first introduces the integral for the ordinate sets of what kind of function? Back: Step functions. 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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Basic
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).
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Basic
Let s be a step function. \int_a^b s(x) \,dx corresponds to what big operator?
Back: \sum
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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Basic
Let s be a step function. What does s_k denote in the following definition? \int_a^b s(x) ,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})
Back: The value of s on the k$th open subinterval of $s's partition.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let s be a step function. What does n denote in the following definition? \int_a^b s(x) ,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})
Back: The number of subintervals in s's partition.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let s be a step function. What does x_k denote in the following definition? \int_a^b s(x) ,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})
Back: The right endpoint of the k$th subinterval of $s's partition.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let s be a step function. What does x_{k-1} denote in the following definition? \int_a^b s(x) ,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})
Back: The left endpoint of the k$th subinterval of $s's partition.
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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Basic
Let s be a step function. What shape is described in the summation body of the following definition? \int_a^b s(x) ,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})
Back: A rectangle.
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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Basic
Let s be a step function. Why don't the values at subdivision points matter in the following definition? \int_a^b s(x) ,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})
Back: The area of a line segment is 0.
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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Basic
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).
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Basic
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).
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%%ANKI Cloze The {ordinate set} of any step function consists of a finite number of {rectangles}. 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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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 = 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).
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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$$
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Basic
What does the additivity property w.r.t. the integrand state?
Back: Let f and g be integrable over [a, b]. Then \int_a^b f(x) + g(x) \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Assume the following integrals are defined. What is the following identity called? \int_a^b f(x) + g(x) ,dx = \int_a^b f(x) ,dx + \int_a^b g(x) ,dx$$
Back: The additive property 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).
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Basic
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$$
Back: As \int_a^b f(x) + g(x) \,dx.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Homogeneousness
Let f be integrable over [a, b] and c \in \mathbb{R}. Then \int_a^b c \cdot f(x) ,dx = c\int_a^b f(x) ,dx$$
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Basic
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).
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Basic
Assume the following integrals are defined. What is the following identity called? \int_a^b c \cdot f(x) ,dx = c\int_a^b f(x) ,dx$$
Back: The homogeneous property. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Linearity
Let f and g be integrable over [a, b]. Let c_1, c_2 \in \mathbb{R}. Then \int_a^b [c_1f(x) + c_2g(x)] ,dx = c_1 \int_a^b f(x) ,dx + c_2 \int_a^b g(x) ,dx$$
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Basic
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$$
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Assume the following integrals are defined. What is the following identity called? \int_a^b [c_1f(x) + c_2g(x)] ,dx = c_1 \int_a^b f(x) ,dx + c_2 \int_a^b g(x) ,dx$$
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).
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%%ANKI Basic The linearity property is immediately derived from what other two 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).
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%%ANKI Cloze 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).
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Comparison Theorem
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$$
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Basic
What does the comparison theorem for integrals state?
Back: Let f and g be integrable over [a, b]. If f(x) \leq g(x) for all x \in [a, b], then \int_a^b f(x) ,dx \leq \int_a^b g(x) ,dx.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let f and g be integrable over [a, b] such that f(x) \leq g(x) for all x \in [a, b]. What is the following called? \int_a^b f(x) ,dx \leq \int_a^b g(x) ,dx$$
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).
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%%ANKI Basic 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).
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%%ANKI Basic 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).
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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$$
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Basic
What does the additivity property w.r.t. the interval of integration state?
Back: If two of the following three integrals exist, the third also exists, and we have \int_a^b f(x) ,dx + \int_b^c f(x) ,dx = \int_a^c f(x) ,dx$$
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Assume the following integrals are defined. What is the following identity called? \int_a^b f(x) ,dx + \int_b^c f(x) ,dx = \int_a^c f(x) ,dx$$
Back: The additive property w.r.t. the interval of integration. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Assume the following integrals exist. How is the following written more compactly? \int_a^b f(x) ,dx + \int_b^c f(x) ,dx$$
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).
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%%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).
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%%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).
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Invariance Under Translation
Let f be integrable over [a, b] and c \in \mathbb{R}. Then \int_a^b f(x) ,dx = \int_{a+c}^{b+c} f(x - c) ,dx$$
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Basic
What does the invariance under translation propery of integrals state?
Back: Let f be integrable over [a, b] and c \in \mathbb{R}. Then \int_a^b f(x) ,dx = \int_{a+c}^{b+c} f(x - c) ,dx$$
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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Basic
Assume the following integrals are defined. What is the following identity called? \int_a^b f(x) ,dx = \int_{a+c}^{b+c} f(x - c) ,dx$$
Back: Invariance under translation.
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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%%ANKI Basic Invariance of integrals under translation corresponds to what property of area? Back: Invariance under congruence. 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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%%ANKI Basic Invariance of area under congruence corresponds to what basic property of integrals? Back: Invariance under translation. 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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Basic
Let f be integrable over [a, b] and g(x) = f(x - c). What integral of g equals \int_a^b f(x) \,dx?
Back: \int_{a+c}^{b+c} 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).
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Basic
Let f be integrable over [a, b] and g(x) = f(x + c). What integral of g equals \int_a^b f(x) \,dx?
Back: \int_{a-c}^{b-c} 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).
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Bibliography
- “Integral.” In Wikipedia, December 31, 2024. https://en.wikipedia.org/w/index.php?title=Integral.
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).