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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Integrals | Obsidian::STEM | calculus::integrals |
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Overview
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
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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Bibliography
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).