--- title: Integrals TARGET DECK: Obsidian::STEM FILE TAGS: calculus::integrals tags: - calculus - integral --- ## Overview 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: $$\int_a^b s(x) \,dx = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1})$$ %%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). END%% %%ANKI 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). 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). 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). 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%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). END%% ## Bibliography * Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).