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Integrals Obsidian::STEM calculus::integral
calculus
integral

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 integration limits. 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.

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI Basic Let f be integrable over [a, b]. What does \int_a^b f(x) \,dx evaluate to after swapping integration limits? 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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.

END%%

%%ANKI 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.

END%%

%%ANKI 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.

END%%

%%ANKI Basic What name is given collectively to a and b in \int_a^b f(x) \,dx? Back: The integration limits. Reference: “Integral.” In Wikipedia, December 31, 2024. https://en.wikipedia.org/w/index.php?title=Integral.

END%%

%%ANKI 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.

END%%

%%ANKI 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.

END%%

%%ANKI 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.

END%%

%%ANKI 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.

END%%

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).

END%%

%%ANKI 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).

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 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).

END%%

%%ANKI 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).

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%%

%%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 = 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).

END%%

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$$

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

Vertical Scaling

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.

This is also known as the homogeneous property.

%%ANKI 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).

END%%

%%ANKI Basic What does the vertical scaling 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).

END%%

%%ANKI 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 vertical scaling (or homogeneous) property. 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 {vertical scaling} property of integrals is also known as 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).

END%%

%%ANKI Cloze The {vertical scaling} property of integrals corresponds to a change of scale on the {y}-axis. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

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$$

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI Basic The linearity property is immediately derived from what other two properties? Back: The additive property w.r.t. the integrand and vertical scaling. 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 {linearity} property of integrals follows from the {additivity w.r.t the integrand} property and the {homogenous} property. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

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$$

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%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).

END%%

%%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).

END%%

Interval of Integration Additivity

Let f be integrable over an interval containing a, b, and c. Then \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: Let f be integrable over an interval containing a, b, and c. Then \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).

END%%

%%ANKI 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).

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).

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).

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).

END%%

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$$

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%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).

END%%

%%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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI Basic How is the following rewritten so that the integrand is just f(x)? \int_a^b f(x + c) ,dx$$ Back: \int_{a+c}^{b+c} f(x) ,dx$$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic How is the following rewritten so that the integrand is just f(x)? \int_a^b f(x - c) ,dx$$ Back: \int_{a-c}^{b-c} f(x) ,dx$$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic How is the following rewritten so that the integration limits are just a and b? \int_{a+c}^{b+c} f(x) ,dx$$ Back: \int_{a}^{b} f(x + c) ,dx$$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic How is the following rewritten so that the integration limits are just a and b? \int_{a-c}^{b-c} f(x) ,dx$$ Back: \int_{a}^{b} f(x - c) ,dx$$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

Horizontal Scaling

Let f be integrable on [a, b]. Then for all k \in \mathbb{R} such that k \neq 0, \int_a^b f(x) ,dx = \frac{1}{k} \int_{ka}^{kb} f\left(\frac{x}{k}\right) ,dx.

%%ANKI Basic What does the horizontal scaling property of integrals state? Back: Let f be integrable over [a, b] and k \in \mathbb{R} s.t. k \neq 0. Then \int_a^b f(x) ,dx = \frac{1}{k} \int_{ka}^{kb} f\left(\frac{x}{k}\right) ,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 Assume the following integrals are defined. What is the following identity called? \int_a^b f(x) ,dx = \frac{1}{k} \int_{ka}^{kb} f\left(\frac{x}{k}\right) ,dx$$ Back: The horizontal scaling property. 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 f be integrable over [a, b] and g(x) = f(x / k). What integral of g equals \int_a^b f(x) \,dx? Back: \frac{1}{k} \int_{ka}^{kb} 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).

END%%

%%ANKI Basic Let f be integrable over [a, b] and g(x) = f(kx). What integral of g equals \int_a^b f(x) \,dx? Back: k \int_{a/k}^{b/k} 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).

END%%

%%ANKI Cloze The {horizontal scaling} property of integrals corresponds to a change of scale on the {x}-axis. 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 identity relates horizontal and vertical scaling of integrals? Back: \int_a^b kf(x) ,dx = \int_{ka}^{kb} f\left(\frac{x}{k}\right) ,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 How is the following rewritten so that the integrand is just f(x)? \int_a^b f(kx) ,dx$$ Back: \frac{1}{k} \int_{ka}^{kb} f(x) ,dx$$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic How is the following rewritten so that the integrand is just f(x)? \int_a^b f\left(\frac{x}{k}\right) ,dx$$ Back: $k \int_{a / k}^{b / k} f(x) \,dx$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic How is the following rewritten so that the integration limits are just a and b? \int_{ka}^{kb} f(x) ,dx$$ Back: $k \int_{a}^{b} f(kx) \,dx$ 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 How is the following rewritten so that the integration limits are just a and b? \int_{a/k}^{b/k} f(x) ,dx$$ Back: \frac{1}{k} \int_{a}^{b} f\left(\frac{x}{k}\right) ,dx$$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

Reflection

The reflection properties are special cases of the vertical and horizontal scaling properties. Let f be integrable over [a, b]. Then \int_a^b f(x) ,dx = -\int_a^b -f(x) ,dx \quad\text{and}\quad \int_a^b f(x) ,dx = \int_{-b}^{-a} f(-x) ,dx.

%%ANKI Basic What does the horizontal reflection property of integrals state? Back: Let f be integrable over [a, b]. Then \int_a^b f(x) ,dx = \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).

END%%

%%ANKI Basic What does the vertical reflection property of integrals state? Back: Let f be integrable over [a, b]. Then \int_a^b f(x) ,dx = -\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).

END%%

%%ANKI Basic Assume the following integrals are defined. What is the following identity called? \int_a^b f(x) ,dx = \int_{-b}^{-a} f(-x) ,dx. Back: The horizontal reflection 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 Assume the following integrals are defined. What is the following identity called? \int_a^b f(x) ,dx = -\int_{a}^{b} -f(x) ,dx. Back: The vertical reflection 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 horizontal reflection property of integrals is a special case of what other basic property? Back: Horizontal scaling. 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 vertical reflection property of integrals is a special case of what other basic property? Back: Vertical scaling. 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 reflection property of integrals is a scaling by what value? Back: -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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