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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.
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})
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? 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
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).
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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).
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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).
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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).
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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 relate to refinements of s's partition?
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 s be a constant function. What does \int_a^b s(x) \,dx evaluate to?
Back: c(b - a)
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_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).
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%%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})$
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
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: As -\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).
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%%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).
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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).
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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).
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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).
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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$$
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Basic
Let s and t be step functions over [a, b]. What does the additive property 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).
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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$$
Back: The additive property. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let s and t be step functions over [a, b]. How is the following more compactly written? \int_a^b s(x) ,dx + \int_a^b t(x) ,dx$$
Back: \int_a^b s(x) + t(x) \,dx
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 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$$
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Basic
Let s be a step function over [a, b]. What does the homogeneous property state?
Back: For all c \in \mathbb{R}, \int_a^b c \cdot s(x) \,dx = c \int_a^b s(x) \,dx.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let s be a step function defined over [a, b] and c \in \mathbb{R}. What is the following identity called? \int_a^b c \cdot s(x) ,dx = c\int_a^b s(x) ,dx$$
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 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$$
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Basic
Let s and t be step functions over [a, b] and c_1, c_2 \in \mathbb{R}. What does the linearity property state?
Back: \int_a^b [c_1 s(x) + c_2 t(x)] \,dx = c_1 \int_a^b s(x) \,dx + c_2 \int_a^b t(x) \,dx
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI
Basic
Let s and t be step functions over [a, b] and c_1, c_2 \in \mathbb{R}. What is the following identity called? \int_a^b [c_1s(x) + c_2t(x)] ,dx = c_1 \int_a^b s(x) ,dx + c_2 \int_a^b t(x) ,dx$$
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 and homogeneous properties. 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 is a combination of the {additive} and {homogenous} properties. 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 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$$
%%ANKI
Basic
Let s and t be step functions over [a, b]. What does the comparison theorem state?
Back: If s(x) < t(x) for all x \in [a, b], \int_a^b s(x) \,dx < \int_a^b t(x) \,dx.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let s and t be step functions over [a, b] such that s(x) < t(x) for all x \in [a, b]. What is the following called? \int_a^b s(x) ,dx < \int_a^b t(x) ,dx$$
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 step function integrals corresponds to what property of area? Back: The monotone 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 monotone property of area corresponds to what theorem of step function 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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Bibliography
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).