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TOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time;FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).TRAPEZOIDAL RULE
We know from a previous lesson that we can use Riemann Sums to evaluate a definite integral \(\int\limits_a^b {f\left( x \right)dx}.\). Riemann Sums use rectangles to approximate the area under a curve. Another useful integration rule is the Trapezoidal Rule.Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. SURFACE INTEGRALS OF SCALAR FUNCTIONS Surface Integrals of Scalar Functions. Consider a scalar function f (x,y,z) and a surface S. Let S be given by the position vector. r(u,v) = x(u,v)i+ y(u,v)j+ z(u,v)k, where the coordinates (u,v) range over some domain D(u,v) of the uv -plane. Notice that the function f (x,y,z) is evaluated only on the points of the surface S, that is.TOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. PROPERTIES OF RELATIONS A binary relation \(R\) defined on a set \(A\) may have the following properties:. Reflexivity; Irreflexivity; Symmetry; Antisymmetry; Asymmetry; Transitivity; Next we will discuss these properties in more detail. Reflexive Relation. A binary relation \(R\) is called reflexive if and only if \(\forall a \in A,\) \(aRa.\) So, a relation \(R\) is reflexive if it relates every element of \(A\) to COMPOSITION OF RELATIONS The composition of R and S, denoted by S ∘R, is a binary relation from A to C, if and only if there is a b ∈ B such that aRb and bSc. Formally the composition S∘ R can be written as. S ∘R = {(a,c) ∣ ∃b ∈ B: aRb∧ bSc}, where a ∈ A and c ∈ C. The composition of binary relations is associative, but not commutative. INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time;TRAPEZOIDAL RULE
We know from a previous lesson that we can use Riemann Sums to evaluate a definite integral \(\int\limits_a^b {f\left( x \right)dx}.\). Riemann Sums use rectangles to approximate the area under a curve. Another useful integration rule is the Trapezoidal Rule.Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x). SURFACE INTEGRALS OF SCALAR FUNCTIONS Surface Integrals of Scalar Functions. Consider a scalar function f (x,y,z) and a surface S. Let S be given by the position vector. r(u,v) = x(u,v)i+ y(u,v)j+ z(u,v)k, where the coordinates (u,v) range over some domain D(u,v) of the uv -plane. Notice that the function f (x,y,z) is evaluated only on the points of the surface S, that is. TRIGONOMETRIC AND HYPERBOLIC SUBSTITUTIONS In this section we consider the integration of functions containing a radical of the form \(\sqrt {a{x^2} + bx + c}.\) When calculating such an integral, we first needHOME PAGE
Over 2000 Solved Problems covering all major topics from Set Theory to Systems of Differential Equations Clear Explanation of Theoretical Concepts makes the website accessible to high school, college and university math students. Step-by-Step Solutions of typical problems that students can encounter while learning mathematics. A Large Variety of Applications See, for example, 20+TOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and.FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. APPLICATIONS OF INTEGRALS Applications of Integrals. In this section, we will take a look at some applications of the definite integral. We will look how to use integrals to calculate volume, surface area, arc length, area between curves, average function value and other mathematical quantities. We will also explore applications of integration in physics andeconomics.
SIMPSON'S RULE
Simpson’s Rule is a numerical method that approximates the value of a definite integral by using quadratic functions.. This method is named after the English mathematician Thomas Simpson \(\left( {1710 – 1761} \right).\) Simpson’s Rule is based on the fact that given three points, we can find the equation of a quadratic through thosepoints.
DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).PAPPUS'S THEOREM
Pappus’s theorem (also known as Pappus’s centroid theorem, Pappus-Guldinus theorem or the Guldinus theorem) deals with the areas of surfaces of revolution and with the volumes of solids of revolution.. The Pappus’s theorem is actually two theorems that allow us to find surface areas and volumes without using integration. Pappus’s Theorem for Surface Area CHANGE OF VARIABLES IN DOUBLE INTEGRALS Thus, use of change of variables in a double integral requires the following 3 steps: Find the pulback S in the new coordinate system (u,v) for the initial region of integration R; Calculate the Jacobian of the transformation (x,y) → (u,v) and write down the differential through the new variables: dxdy = ∣∣ ∣ EQUATION OF CATENARY So it was believed for a long time. In the early \(17\)th century Galileo doubted that a hanging chain is actually a parabola. However, a rigorous proof was obtained only half a century later after Isaak Newton and Gottfried Leibniz developed a framework of differential and integral calculus.. The solution of the problem about the catenary was published in \(1691\) by Christiaan Huygens PROPERTIES OF RELATIONS A binary relation \(R\) defined on a set \(A\) may have the following properties:. Reflexivity; Irreflexivity; Symmetry; Antisymmetry; Asymmetry; Transitivity; Next we will discuss these properties in more detail. Reflexive Relation. A binary relation \(R\) is called reflexive if and only if \(\forall a \in A,\) \(aRa.\) So, a relation \(R\) is reflexive if it relates every element of \(A\) toTOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time;FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).TRAPEZOIDAL RULE
We know from a previous lesson that we can use Riemann Sums to evaluate a definite integral \(\int\limits_a^b {f\left( x \right)dx}.\). Riemann Sums use rectangles to approximate the area under a curve. Another useful integration rule is the Trapezoidal Rule.Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. SURFACE INTEGRALS OF SCALAR FUNCTIONS Surface Integrals of Scalar Functions. Consider a scalar function f (x,y,z) and a surface S. Let S be given by the position vector. r(u,v) = x(u,v)i+ y(u,v)j+ z(u,v)k, where the coordinates (u,v) range over some domain D(u,v) of the uv -plane. Notice that the function f (x,y,z) is evaluated only on the points of the surface S, that is. PROPERTIES OF RELATIONS A binary relation \(R\) defined on a set \(A\) may have the following properties:. Reflexivity; Irreflexivity; Symmetry; Antisymmetry; Asymmetry; Transitivity; Next we will discuss these properties in more detail. Reflexive Relation. A binary relation \(R\) is called reflexive if and only if \(\forall a \in A,\) \(aRa.\) So, a relation \(R\) is reflexive if it relates every element of \(A\) toTOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time;FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).TRAPEZOIDAL RULE
We know from a previous lesson that we can use Riemann Sums to evaluate a definite integral \(\int\limits_a^b {f\left( x \right)dx}.\). Riemann Sums use rectangles to approximate the area under a curve. Another useful integration rule is the Trapezoidal Rule.Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. SURFACE INTEGRALS OF SCALAR FUNCTIONS Surface Integrals of Scalar Functions. Consider a scalar function f (x,y,z) and a surface S. Let S be given by the position vector. r(u,v) = x(u,v)i+ y(u,v)j+ z(u,v)k, where the coordinates (u,v) range over some domain D(u,v) of the uv -plane. Notice that the function f (x,y,z) is evaluated only on the points of the surface S, that is.HOME PAGE
Over 2000 Solved Problems covering all major topics from Set Theory to Systems of Differential Equations Clear Explanation of Theoretical Concepts makes the website accessible to high school, college and university math students. Step-by-Step Solutions of typical problems that students can encounter while learning mathematics. A Large Variety of Applications See, for example, 20+TOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and.FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. APPLICATIONS OF INTEGRALS Applications of Integrals. In this section, we will take a look at some applications of the definite integral. We will look how to use integrals to calculate volume, surface area, arc length, area between curves, average function value and other mathematical quantities. We will also explore applications of integration in physics andeconomics.
SIMPSON'S RULE
Simpson’s Rule is a numerical method that approximates the value of a definite integral by using quadratic functions.. This method is named after the English mathematician Thomas Simpson \(\left( {1710 – 1761} \right).\) Simpson’s Rule is based on the fact that given three points, we can find the equation of a quadratic through thosepoints.
DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).PAPPUS'S THEOREM
Pappus’s theorem (also known as Pappus’s centroid theorem, Pappus-Guldinus theorem or the Guldinus theorem) deals with the areas of surfaces of revolution and with the volumes of solids of revolution.. The Pappus’s theorem is actually two theorems that allow us to find surface areas and volumes without using integration. Pappus’s Theorem for Surface Area CHANGE OF VARIABLES IN DOUBLE INTEGRALS Thus, use of change of variables in a double integral requires the following 3 steps: Find the pulback S in the new coordinate system (u,v) for the initial region of integration R; Calculate the Jacobian of the transformation (x,y) → (u,v) and write down the differential through the new variables: dxdy = ∣∣ ∣ EQUATION OF CATENARY So it was believed for a long time. In the early \(17\)th century Galileo doubted that a hanging chain is actually a parabola. However, a rigorous proof was obtained only half a century later after Isaak Newton and Gottfried Leibniz developed a framework of differential and integral calculus.. The solution of the problem about the catenary was published in \(1691\) by Christiaan Huygens PROPERTIES OF RELATIONS A binary relation \(R\) defined on a set \(A\) may have the following properties:. Reflexivity; Irreflexivity; Symmetry; Antisymmetry; Asymmetry; Transitivity; Next we will discuss these properties in more detail. Reflexive Relation. A binary relation \(R\) is called reflexive if and only if \(\forall a \in A,\) \(aRa.\) So, a relation \(R\) is reflexive if it relates every element of \(A\) toTOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time;FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).TRAPEZOIDAL RULE
We know from a previous lesson that we can use Riemann Sums to evaluate a definite integral \(\int\limits_a^b {f\left( x \right)dx}.\). Riemann Sums use rectangles to approximate the area under a curve. Another useful integration rule is the Trapezoidal Rule.Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. SURFACE INTEGRALS OF SCALAR FUNCTIONS Surface Integrals of Scalar Functions. Consider a scalar function f (x,y,z) and a surface S. Let S be given by the position vector. r(u,v) = x(u,v)i+ y(u,v)j+ z(u,v)k, where the coordinates (u,v) range over some domain D(u,v) of the uv -plane. Notice that the function f (x,y,z) is evaluated only on the points of the surface S, that is. PROPERTIES OF RELATIONS A binary relation \(R\) defined on a set \(A\) may have the following properties:. Reflexivity; Irreflexivity; Symmetry; Antisymmetry; Asymmetry; Transitivity; Next we will discuss these properties in more detail. Reflexive Relation. A binary relation \(R\) is called reflexive if and only if \(\forall a \in A,\) \(aRa.\) So, a relation \(R\) is reflexive if it relates every element of \(A\) toTOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time;FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).TRAPEZOIDAL RULE
We know from a previous lesson that we can use Riemann Sums to evaluate a definite integral \(\int\limits_a^b {f\left( x \right)dx}.\). Riemann Sums use rectangles to approximate the area under a curve. Another useful integration rule is the Trapezoidal Rule.Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. SURFACE INTEGRALS OF SCALAR FUNCTIONS Surface Integrals of Scalar Functions. Consider a scalar function f (x,y,z) and a surface S. Let S be given by the position vector. r(u,v) = x(u,v)i+ y(u,v)j+ z(u,v)k, where the coordinates (u,v) range over some domain D(u,v) of the uv -plane. Notice that the function f (x,y,z) is evaluated only on the points of the surface S, that is.HOME PAGE
Over 2000 Solved Problems covering all major topics from Set Theory to Systems of Differential Equations Clear Explanation of Theoretical Concepts makes the website accessible to high school, college and university math students. Step-by-Step Solutions of typical problems that students can encounter while learning mathematics. A Large Variety of Applications See, for example, 20+TOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and.FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
BAROMETRIC FORMULA
Solution. The air pressure in the mine can be estimated using the general barometric formula: P = P 0exp(− M g RT h). We substitute the following values into the formula: h = −1000m (the sign is minus because the mine is under sea level); T = 40 +273.15 = 313.15K. APPLICATIONS OF INTEGRALS Applications of Integrals. In this section, we will take a look at some applications of the definite integral. We will look how to use integrals to calculate volume, surface area, arc length, area between curves, average function value and other mathematical quantities. We will also explore applications of integration in physics andeconomics.
SIMPSON'S RULE
Simpson’s Rule is a numerical method that approximates the value of a definite integral by using quadratic functions.. This method is named after the English mathematician Thomas Simpson \(\left( {1710 – 1761} \right).\) Simpson’s Rule is based on the fact that given three points, we can find the equation of a quadratic through thosepoints.
DEFINITION AND PROPERTIES OF THE DERIVATIVE Definition and Properties of the Derivative. The derivative of a function y = f (x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y = f (x) has an increment Δx. Then the increment of the function is equal to. Δy = f (x +Δx) − f (x).PAPPUS'S THEOREM
Pappus’s theorem (also known as Pappus’s centroid theorem, Pappus-Guldinus theorem or the Guldinus theorem) deals with the areas of surfaces of revolution and with the volumes of solids of revolution.. The Pappus’s theorem is actually two theorems that allow us to find surface areas and volumes without using integration. Pappus’s Theorem for Surface Area CHANGE OF VARIABLES IN DOUBLE INTEGRALS Thus, use of change of variables in a double integral requires the following 3 steps: Find the pulback S in the new coordinate system (u,v) for the initial region of integration R; Calculate the Jacobian of the transformation (x,y) → (u,v) and write down the differential through the new variables: dxdy = ∣∣ ∣ EQUATION OF CATENARY So it was believed for a long time. In the early \(17\)th century Galileo doubted that a hanging chain is actually a parabola. However, a rigorous proof was obtained only half a century later after Isaak Newton and Gottfried Leibniz developed a framework of differential and integral calculus.. The solution of the problem about the catenary was published in \(1691\) by Christiaan HuygensTOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. PROPERTIES OF RELATIONS A binary relation \(R\) defined on a set \(A\) may have the following properties:. Reflexivity; Irreflexivity; Symmetry; Antisymmetry; Asymmetry; Transitivity; Next we will discuss these properties in more detail. Reflexive Relation. A binary relation \(R\) is called reflexive if and only if \(\forall a \in A,\) \(aRa.\) So, a relation \(R\) is reflexive if it relates every element of \(A\) toCOUNTING FUNCTIONS
Counting Bijective Functions. If there is a bijection between two finite sets A and B, then the two sets have the same number of elements, that is, |A| = |B| = n. The number of bijective functionsbetween the sets is
INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 COMPOSITION OF RELATIONS The composition of R and S, denoted by S ∘R, is a binary relation from A to C, if and only if there is a b ∈ B such that aRb and bSc. Formally the composition S∘ R can be written as. S ∘R = {(a,c) ∣ ∃b ∈ B: aRb∧ bSc}, where a ∈ A and c ∈ C. The composition of binary relations is associative, but not commutative. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time; SET OPERATIONS AND VENN DIAGRAMS Sets are treated as mathematical objects. Similarly to numbers, we can perform certain mathematical operations on sets.Below we consider the principal operations involving the intersection, union, difference, symmetric difference, and the complement of sets.. To visualize set operations, we will use Venn diagrams.In a Venn diagram, a rectangle shows the universal set, and all other sets are OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits.FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
RICCATI EQUATION
The Riccati equation is one of the most interesting nonlinear differential equations of first order. It’s written in the form: y′ = a(x)y+ b(x)y2 +c(x), where a(x), b(x), c(x) are continuous functions of x. The Riccati equation is used in different areas of mathematics (for example, in algebraic geometry and the theory ofconformal mapping
TOTAL ORDERS
A partially ordered set (A,≼) in which any two elements are comparable is called a total order. Total orders are also sometimes called linear orders. Formally, a binary relation R on a non-empty set A is a total order if the relation is. connex. antisymmetric, and. PROPERTIES OF RELATIONS A binary relation \(R\) defined on a set \(A\) may have the following properties:. Reflexivity; Irreflexivity; Symmetry; Antisymmetry; Asymmetry; Transitivity; Next we will discuss these properties in more detail. Reflexive Relation. A binary relation \(R\) is called reflexive if and only if \(\forall a \in A,\) \(aRa.\) So, a relation \(R\) is reflexive if it relates every element of \(A\) toCOUNTING FUNCTIONS
Counting Bijective Functions. If there is a bijection between two finite sets A and B, then the two sets have the same number of elements, that is, |A| = |B| = n. The number of bijective functionsbetween the sets is
INTEGRATION BY COMPLETING THE SQUARE Completing the square helps when quadratic functions are involved in the integrand. When the integrand is a rational function with a quadratic expression in the denominator, we can use the following table integrals: 1. ∫ dx √x2 ±a2 = ln∣∣x+√x2 ±a2∣∣. 2. ∫ dx √a2 −x2 = arcsin x a. 3. ∫ dx a2 +x2 = 1 a arctan x a. 4 COMPOSITION OF RELATIONS The composition of R and S, denoted by S ∘R, is a binary relation from A to C, if and only if there is a b ∈ B such that aRb and bSc. Formally the composition S∘ R can be written as. S ∘R = {(a,c) ∣ ∃b ∈ B: aRb∧ bSc}, where a ∈ A and c ∈ C. The composition of binary relations is associative, but not commutative. APPLICATIONS OF INTEGRALS IN ECONOMICS Applications of Integrals in Economics. The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time; SET OPERATIONS AND VENN DIAGRAMS Sets are treated as mathematical objects. Similarly to numbers, we can perform certain mathematical operations on sets.Below we consider the principal operations involving the intersection, union, difference, symmetric difference, and the complement of sets.. To visualize set operations, we will use Venn diagrams.In a Venn diagram, a rectangle shows the universal set, and all other sets are OPTIMIZATION PROBLEMS IN ECONOMICS Optimization Problems in Economics. In business and economics there are many applied problems that require optimization. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits.FLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
RICCATI EQUATION
The Riccati equation is one of the most interesting nonlinear differential equations of first order. It’s written in the form: y′ = a(x)y+ b(x)y2 +c(x), where a(x), b(x), c(x) are continuous functions of x. The Riccati equation is used in different areas of mathematics (for example, in algebraic geometry and the theory ofconformal mapping
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Over 2000 Solved Problems covering all major topics from Set Theory to Systems of Differential Equations Clear Explanation of Theoretical Concepts makes the website accessible to high school, college and university math students. Step-by-Step Solutions of typical problems that students can encounter while learning mathematics. A Large Variety of Applications See, for example, 20+COUNTING FUNCTIONS
Counting Bijective Functions. If there is a bijection between two finite sets A and B, then the two sets have the same number of elements, that is, |A| = |B| = n. The number of bijective functionsbetween the sets is
RICCATI EQUATION
The Riccati equation is one of the most interesting nonlinear differential equations of first order. It’s written in the form: y′ = a(x)y+ b(x)y2 +c(x), where a(x), b(x), c(x) are continuous functions of x. The Riccati equation is used in different areas of mathematics (for example, in algebraic geometry and the theory ofconformal mapping
SET OPERATIONS AND VENN DIAGRAMS Sets are treated as mathematical objects. Similarly to numbers, we can perform certain mathematical operations on sets.Below we consider the principal operations involving the intersection, union, difference, symmetric difference, and the complement of sets.. To visualize set operations, we will use Venn diagrams.In a Venn diagram, a rectangle shows the universal set, and all other sets areFLUID PRESSURE
Fluid Pressure. Pressure is defined as the force per unit area: P = F A. If an object is immersed in a liquid at a depth h, the fluid pressure is given by the constant depth formula. where ρ is the fluid density and g is the acceleration due to gravity. Fluid pressure is ascalar quantity.
LOGIC AND SET NOTATION Set theory is a branch of mathematical logic.Therefore, it is natural that logical language and symbols are used to describe sets. In this section, we will look at the basic logical symbols and ways ofdefining sets.
EQUIVALENCE CLASSES AND PARTITIONS There is a direct link between equivalence classes and partitions. For any equivalence relation on a set A, the set of all its equivalence classes is a partition of A. The converse is also true. Given a partition P on set A, we can define an equivalence relation induced by the partition such that a ∼ b if and only if the elements a and bare
COMPLEX FORM OF FOURIER SERIES Using the well-known Euler’s formulas. we can write the Fourier series of the function in complex form: f (x) = a0 2 + ∞ ∑ n=1(ancosnx+bnsinnx) = a0 2 + ∞ ∑ n=1(an einx +e−inx 2 + bn einx −e−inx 2i) = a0 2 + ∞ ∑ n=1 an −ibn 2 einx + ∞ ∑ n=1 an + ibn 2 e−inx = ∞ ∑ n=−∞cneinx. Here we have used thefollowing
EVEN AND ODD EXTENSIONS This can be done in two ways: We can construct the even extension of f (x): f even (x) = {f (−x), −π ≤ x < 0 f (x), 0 ≤ x ≤ π, or the odd extension of f (x): f odd(x) = {−f (−x), −π ≤ x < 0 f (x), 0 ≤ x ≤ π. For the even function, the Fourier series is called the Fourier Cosine series and is given by. f even (x) = a0 2 CHANGE OF VARIABLES IN DOUBLE INTEGRALS Thus, use of change of variables in a double integral requires the following 3 steps: Find the pulback S in the new coordinate system (u,v) for the initial region of integration R; Calculate the Jacobian of the transformation (x,y) → (u,v) and write down the differential through the new variables: dxdy = ∣∣ ∣Skip to content
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