The document discusses the concept of approximating the area under a curve using rectangles. It shows that as the width of the rectangles decreases, the approximation becomes more accurate. The area under the curve between two points can be estimated as the difference in the area functions at those points. Ultimately, the derivative of the area function gives the exact equation of the curve, and the total area under the curve between two x-values can be calculated as the difference in the area function evaluated at those x-values.
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The document discusses Pappus's centroid theorem, which states that the volume of a solid generated by rotating a region R about a line l is equal to the product of the area of R and the distance traveled by R's centroid during one full rotation. The theorem can be used to calculate the volumes and surface areas of various solids of revolution. Examples are provided of applying the theorem to find volumes and surface areas of solids like cones, spheres, and others.
The document provides steps to sketch the path of a rock launched from a catapult using trigonometric functions. It determines that the path will be a negative cosine curve. It then finds the parameters for the cosine equation f(x) = -90cos(x/50) + 90 based on the given maximum height and range of the catapult. Finally, it derives the equivalent sine equation f(x) = -90sin(x+22.5)+90.
The document provides steps to sketch the path of a rock launched from a catapult based on given parameters. It determines the cosine equation f(x) = -90 cos(x/50) + 90 that models the rock's maximum height of 180m and range of 100m, and the equivalent sine equation f(x) = 90 sin(x/50 - 22.5) + 90.
The document discusses the aerodynamic drag of bodies of revolution, specifically fuselages. It describes drag as having pressure and friction components. At supersonic speeds, wave drag from pressure disturbances also occurs. Drag is divided into zero-lift drag and induced drag related to lift. The wave drag of a fuselage can be estimated by calculating the drag of the nose and rear sections. The nose drag depends on the nose shape and Mach number. The rear drag depends on the tapering, aspect ratio, and Mach number. Engineering methods exist to estimate wave drag using local cones for the nose shape.
The document discusses various methods for calculating the volume of solids of revolution using integration, including the disk method, washer method, and cylindrical shell method. It provides examples of applying each method to find the volume of solids formed by revolving regions between curves about axes. The disk method approximates the solid as disks of thickness dx or dy. The washer method accounts for solids with holes by using washers of thickness dx or dy and inner and outer radii. Examples demonstrate setting up and evaluating the integrals to find the volumes.
This document discusses methods for calculating volumes using integration. There are three main methods - disc, washer, and shell. The disc and washer methods use the formula πr2h to find the volume of revolution around an axis, with disc used for hollow shapes and washer for shapes within other shapes. The shell method uses 2πrh and is primarily used for revolutions around the y-axis. It is important to determine the correct graph and limits of integration based on the problem.
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The document discusses Pappus's centroid theorem, which states that the volume of a solid generated by rotating a region R about a line l is equal to the product of the area of R and the distance traveled by R's centroid during one full rotation. The theorem can be used to calculate the volumes and surface areas of various solids of revolution. Examples are provided of applying the theorem to find volumes and surface areas of solids like cones, spheres, and others.
The document provides steps to sketch the path of a rock launched from a catapult using trigonometric functions. It determines that the path will be a negative cosine curve. It then finds the parameters for the cosine equation f(x) = -90cos(x/50) + 90 based on the given maximum height and range of the catapult. Finally, it derives the equivalent sine equation f(x) = -90sin(x+22.5)+90.
The document provides steps to sketch the path of a rock launched from a catapult based on given parameters. It determines the cosine equation f(x) = -90 cos(x/50) + 90 that models the rock's maximum height of 180m and range of 100m, and the equivalent sine equation f(x) = 90 sin(x/50 - 22.5) + 90.
The document discusses the aerodynamic drag of bodies of revolution, specifically fuselages. It describes drag as having pressure and friction components. At supersonic speeds, wave drag from pressure disturbances also occurs. Drag is divided into zero-lift drag and induced drag related to lift. The wave drag of a fuselage can be estimated by calculating the drag of the nose and rear sections. The nose drag depends on the nose shape and Mach number. The rear drag depends on the tapering, aspect ratio, and Mach number. Engineering methods exist to estimate wave drag using local cones for the nose shape.
The document discusses various methods for calculating the volume of solids of revolution using integration, including the disk method, washer method, and cylindrical shell method. It provides examples of applying each method to find the volume of solids formed by revolving regions between curves about axes. The disk method approximates the solid as disks of thickness dx or dy. The washer method accounts for solids with holes by using washers of thickness dx or dy and inner and outer radii. Examples demonstrate setting up and evaluating the integrals to find the volumes.
This document discusses methods for calculating volumes using integration. There are three main methods - disc, washer, and shell. The disc and washer methods use the formula πr2h to find the volume of revolution around an axis, with disc used for hollow shapes and washer for shapes within other shapes. The shell method uses 2πrh and is primarily used for revolutions around the y-axis. It is important to determine the correct graph and limits of integration based on the problem.
1. The document discusses calculating the area under curves using integral calculus. It provides examples of finding the area bounded by curves like circles, ellipses, parabolas, and lines using integrals.
2. The key approach discussed is treating the area under a curve as the limit of thin vertical or horizontal strips, and expressing the total area as the integral of the curve's equation between the bounds. Both vertical and horizontal strip approaches are demonstrated.
3. Several examples are provided of calculating areas bounded by standard curves like circles and ellipses using integrals, as well as areas between curves and lines. The document concludes with exercises for readers to practice similar area calculations.
This document provides an overview of calculating volumes of solids using the disk and washer methods. It defines key concepts like using the disk method when the cross-sections are disks and the washer method when there is a hole. Examples are given of how to present these methods using a cucumber and cantaloupe. Remediation resources like videos are suggested to help with common issues like determining radii and limits of integration. Finally, several past AP exam questions involving volumes of revolution are referenced.
This document provides an overview of different integration methods for calculating volumes rotated around axes:
1) The Washer Method calculates volumes between two curves rotated around an axis by integrating the area of circular cross-sections.
2) The Inverse Area Method finds areas between curves by rewriting the functions in inverse form and integrating the top curve minus the bottom curve.
3) The Shell Method, similar to Washer, switches the variable of integration depending on the axis of rotation and uses the thickness and height of shell-shaped cross-sections.
4) Other methods discussed include Left Rectangular Approximation and the Disk Method for calculating the volume of a single rotated function using circular cross-sections.
The document discusses trigonometric graphs and identities. It covers graphing sine and cosine functions including amplitude, period, translations involving phase shift and vertical shift. It also provides an example of using trigonometric functions to model real-life tidal data and finds the depth at specific times as well as time periods when a boat can safely dock.
The document discusses polar coordinates and graphs. Polar coordinates (r, θ) can be used to specify the location of a point P by giving the distance r from the origin and the angle θ. Conversion formulas allow changing between polar (r, θ) and rectangular (x, y) coordinates. Polar equations relate r and θ, and common ones like r = c (a circle) and θ = c (a line) are examined. Graphing polar equations involves plotting the r and θ values specified by the equation.
The document discusses parametric equations and parametric curves. Parametric equations define curves using two equations, where x and y are defined in terms of a parameter t. Common examples of parametric curves are given. Parametric equations can be used to define circles by giving the x and y coordinates in terms of the center point and radius. The parameter t can be eliminated to obtain the Cartesian equation of the curve. Calculus can be applied to parametric curves to find tangents, areas, arc length, and surfaces of revolution. Formulas are given for finding the equations of tangent lines to parametric curves at a given value of t.
Polar coordinates provide an alternative way to specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the x-axis and a line from O to P. θ is measured counter-clockwise from the x-axis and can be either positive or negative. A point P's polar coordinates (r, θ) uniquely identify its location. Polar and rectangular coordinates can be converted between each using the relationships x=r*cos(θ), y=r*sin(θ), and r=√(x2+y2).
The document discusses calculating the area swept out by a polar function r=f(θ) between two angles θ=A and θ=B. It introduces the integral formula for finding this area, which is ∫f(θ)2dθ from A to B. It then provides examples of using this formula to calculate the areas of different polar curves, such as a circle, a cardioid, and a curve tracing a circle twice.
The document discusses calculus concepts related to parametric curves including calculating tangents, areas, arc length, and surfaces of revolution. It provides the formulas for finding the tangent of a parametric curve, area under a parametric curve, arc length of a parametric curve, and surface area obtained by rotating a parametric curve around an axis. Examples are given for each concept using the parametric cycloid curve.
1. The document discusses reflection of light from plane and curved mirrors. It defines key terms used to describe reflection such as focal length, radius of curvature, and magnification.
2. Rules for image formation by curved mirrors are presented, including the location, size, and nature of the image for different positions of the object in front of concave and convex mirrors.
3. Examples with solutions are provided to illustrate concepts such as reflection angles, number of images formed by inclined mirrors, and speeds of moving mirrors and images.
1. The document discusses several methods for calculating the volume of solids of revolution, including slicing, disk, washer, and specific examples of their use.
2. The slicing method is used when the cross sections are all the same regular shape. The disk and washer methods are used when revolving an area about an axis, with disk for solids without hollow parts and washer for solids with hollow parts.
3. Examples show how to set up the integrals to calculate volume using the appropriate method, finding radii of cross sections and limits of integration.
This document discusses using the method of cylindrical shells to calculate volumes of solids of revolution. It provides an example calculating the volume of the solid obtained by rotating the region between the curves y=2x^2 - x^3 and y=0 about the y-axis. The method involves imagining the solid as being composed of cylindrical shells and using the formula V=2π∫_{a}^{b} x*f(x) dx to calculate the volume, where f(x) is the height of each shell.
35 tangent and arc length in polar coordinatesmath266
The document discusses parametric representations of polar curves. It begins by explaining that a rectangular curve given by y=f(x) can be parametrized as x=t and y=f(t). It then explains that a polar curve given by r=f(θ) can be parametrized as x=f(θ)cos(θ) and y=f(θ)sin(θ). Examples are given of parametrizing the Archimedean spiral r=θ and the cardioid r=1+sin(θ). Formulas are derived for calculating the slope of the tangent line to a polar curve at a given point in terms of r and θ. Examples are worked out for finding the slope
Materi kuliah tentang Aplikasi Integral. Cari lebih banyak mata kuliah Semester 1 di: http://muhammadhabibielecture.blogspot.com/2014/12/kuliah-semester-1-thp-ftp-ub.html
The document discusses parametric equations, which describe the motion of a particle in a plane by giving its position (x, y) at time t as functions of t, known as parametric equations. Examples are provided of parametric equations defining circles, ellipses, and other curves. The parameter does not need to be time. Slopes of parametric curves can be found from the derivatives of the parametric equations with respect to t. Standard parameterizations of functions y=f(x) are also discussed.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
www.anuragtyagiclasses.com
Este campo trata de un conjunto de puntos del plano que equidistan una unidad del origen; la distancia de tales puntos al centro se denomina radio de circunferencia.
The document discusses vector calculus concepts including:
1) Coordinate systems used in vector calculus problems including rectangular, cylindrical, and spherical coordinates.
2) How to write vectors and their components in each coordinate system.
3) Relationships between vectors in different coordinate systems using transformation matrices.
4) Concepts of gradient, divergence, and curl and their definitions and representations in different coordinate systems.
5) Theorems relating integrals, including the divergence theorem and Stokes' theorem.
This document provides formulas for calculating the centroids and areas of various geometric shapes including triangles, circles, semicircles, quarter circles, ellipses, parabolas, arcs, and sectors. The formulas give the x and y coordinates of the centroid and expressions for calculating the area of each shape based on parameters like side lengths, radii, angles, etc. Formulas are provided for basic as well as composite shapes formed by combining geometric elements.
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses how to calculate the area under a curve using definite integrals and the Fundamental Theorem of Calculus. It explains that the area can be approximated as the sum of rectangles and becomes exact as the width approaches zero. The area is then given by the definite integral from a to b of the function, which is equal to evaluating the antiderivative at b and subtracting the evaluation at a. Examples demonstrate calculating areas under parabolic and exponential curves using this process.
This document discusses calculating the area under a curve using integration. It begins by approximating the area under an irregular shape using squares and rectangles. It then introduces defining the area A as a limit of approximating rectangles as their width approaches 0. This is written as the integral from a to b of f(x) dx, where f(x) is the curve. Examples are given of setting up definite integrals to calculate the areas under curves and between two curves. Steps for determining the area of a plane figure using integration are also provided.
1. The document discusses calculating the area under curves using integral calculus. It provides examples of finding the area bounded by curves like circles, ellipses, parabolas, and lines using integrals.
2. The key approach discussed is treating the area under a curve as the limit of thin vertical or horizontal strips, and expressing the total area as the integral of the curve's equation between the bounds. Both vertical and horizontal strip approaches are demonstrated.
3. Several examples are provided of calculating areas bounded by standard curves like circles and ellipses using integrals, as well as areas between curves and lines. The document concludes with exercises for readers to practice similar area calculations.
This document provides an overview of calculating volumes of solids using the disk and washer methods. It defines key concepts like using the disk method when the cross-sections are disks and the washer method when there is a hole. Examples are given of how to present these methods using a cucumber and cantaloupe. Remediation resources like videos are suggested to help with common issues like determining radii and limits of integration. Finally, several past AP exam questions involving volumes of revolution are referenced.
This document provides an overview of different integration methods for calculating volumes rotated around axes:
1) The Washer Method calculates volumes between two curves rotated around an axis by integrating the area of circular cross-sections.
2) The Inverse Area Method finds areas between curves by rewriting the functions in inverse form and integrating the top curve minus the bottom curve.
3) The Shell Method, similar to Washer, switches the variable of integration depending on the axis of rotation and uses the thickness and height of shell-shaped cross-sections.
4) Other methods discussed include Left Rectangular Approximation and the Disk Method for calculating the volume of a single rotated function using circular cross-sections.
The document discusses trigonometric graphs and identities. It covers graphing sine and cosine functions including amplitude, period, translations involving phase shift and vertical shift. It also provides an example of using trigonometric functions to model real-life tidal data and finds the depth at specific times as well as time periods when a boat can safely dock.
The document discusses polar coordinates and graphs. Polar coordinates (r, θ) can be used to specify the location of a point P by giving the distance r from the origin and the angle θ. Conversion formulas allow changing between polar (r, θ) and rectangular (x, y) coordinates. Polar equations relate r and θ, and common ones like r = c (a circle) and θ = c (a line) are examined. Graphing polar equations involves plotting the r and θ values specified by the equation.
The document discusses parametric equations and parametric curves. Parametric equations define curves using two equations, where x and y are defined in terms of a parameter t. Common examples of parametric curves are given. Parametric equations can be used to define circles by giving the x and y coordinates in terms of the center point and radius. The parameter t can be eliminated to obtain the Cartesian equation of the curve. Calculus can be applied to parametric curves to find tangents, areas, arc length, and surfaces of revolution. Formulas are given for finding the equations of tangent lines to parametric curves at a given value of t.
Polar coordinates provide an alternative way to specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the x-axis and a line from O to P. θ is measured counter-clockwise from the x-axis and can be either positive or negative. A point P's polar coordinates (r, θ) uniquely identify its location. Polar and rectangular coordinates can be converted between each using the relationships x=r*cos(θ), y=r*sin(θ), and r=√(x2+y2).
The document discusses calculating the area swept out by a polar function r=f(θ) between two angles θ=A and θ=B. It introduces the integral formula for finding this area, which is ∫f(θ)2dθ from A to B. It then provides examples of using this formula to calculate the areas of different polar curves, such as a circle, a cardioid, and a curve tracing a circle twice.
The document discusses calculus concepts related to parametric curves including calculating tangents, areas, arc length, and surfaces of revolution. It provides the formulas for finding the tangent of a parametric curve, area under a parametric curve, arc length of a parametric curve, and surface area obtained by rotating a parametric curve around an axis. Examples are given for each concept using the parametric cycloid curve.
1. The document discusses reflection of light from plane and curved mirrors. It defines key terms used to describe reflection such as focal length, radius of curvature, and magnification.
2. Rules for image formation by curved mirrors are presented, including the location, size, and nature of the image for different positions of the object in front of concave and convex mirrors.
3. Examples with solutions are provided to illustrate concepts such as reflection angles, number of images formed by inclined mirrors, and speeds of moving mirrors and images.
1. The document discusses several methods for calculating the volume of solids of revolution, including slicing, disk, washer, and specific examples of their use.
2. The slicing method is used when the cross sections are all the same regular shape. The disk and washer methods are used when revolving an area about an axis, with disk for solids without hollow parts and washer for solids with hollow parts.
3. Examples show how to set up the integrals to calculate volume using the appropriate method, finding radii of cross sections and limits of integration.
This document discusses using the method of cylindrical shells to calculate volumes of solids of revolution. It provides an example calculating the volume of the solid obtained by rotating the region between the curves y=2x^2 - x^3 and y=0 about the y-axis. The method involves imagining the solid as being composed of cylindrical shells and using the formula V=2π∫_{a}^{b} x*f(x) dx to calculate the volume, where f(x) is the height of each shell.
35 tangent and arc length in polar coordinatesmath266
The document discusses parametric representations of polar curves. It begins by explaining that a rectangular curve given by y=f(x) can be parametrized as x=t and y=f(t). It then explains that a polar curve given by r=f(θ) can be parametrized as x=f(θ)cos(θ) and y=f(θ)sin(θ). Examples are given of parametrizing the Archimedean spiral r=θ and the cardioid r=1+sin(θ). Formulas are derived for calculating the slope of the tangent line to a polar curve at a given point in terms of r and θ. Examples are worked out for finding the slope
Materi kuliah tentang Aplikasi Integral. Cari lebih banyak mata kuliah Semester 1 di: http://muhammadhabibielecture.blogspot.com/2014/12/kuliah-semester-1-thp-ftp-ub.html
The document discusses parametric equations, which describe the motion of a particle in a plane by giving its position (x, y) at time t as functions of t, known as parametric equations. Examples are provided of parametric equations defining circles, ellipses, and other curves. The parameter does not need to be time. Slopes of parametric curves can be found from the derivatives of the parametric equations with respect to t. Standard parameterizations of functions y=f(x) are also discussed.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
www.anuragtyagiclasses.com
Este campo trata de un conjunto de puntos del plano que equidistan una unidad del origen; la distancia de tales puntos al centro se denomina radio de circunferencia.
The document discusses vector calculus concepts including:
1) Coordinate systems used in vector calculus problems including rectangular, cylindrical, and spherical coordinates.
2) How to write vectors and their components in each coordinate system.
3) Relationships between vectors in different coordinate systems using transformation matrices.
4) Concepts of gradient, divergence, and curl and their definitions and representations in different coordinate systems.
5) Theorems relating integrals, including the divergence theorem and Stokes' theorem.
This document provides formulas for calculating the centroids and areas of various geometric shapes including triangles, circles, semicircles, quarter circles, ellipses, parabolas, arcs, and sectors. The formulas give the x and y coordinates of the centroid and expressions for calculating the area of each shape based on parameters like side lengths, radii, angles, etc. Formulas are provided for basic as well as composite shapes formed by combining geometric elements.
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses how to calculate the area under a curve using definite integrals and the Fundamental Theorem of Calculus. It explains that the area can be approximated as the sum of rectangles and becomes exact as the width approaches zero. The area is then given by the definite integral from a to b of the function, which is equal to evaluating the antiderivative at b and subtracting the evaluation at a. Examples demonstrate calculating areas under parabolic and exponential curves using this process.
This document discusses calculating the area under a curve using integration. It begins by approximating the area under an irregular shape using squares and rectangles. It then introduces defining the area A as a limit of approximating rectangles as their width approaches 0. This is written as the integral from a to b of f(x) dx, where f(x) is the curve. Examples are given of setting up definite integrals to calculate the areas under curves and between two curves. Steps for determining the area of a plane figure using integration are also provided.
The document discusses calculating volumes of revolution by rotating an area about the x-axis or y-axis. It provides the formulas for finding these volumes using integration, with examples of setting up the integrals to calculate specific volumes. It also covers cases where the curve needs to be rearranged in order to substitute it into the integral when rotating about the y-axis.
The document discusses using integrals to calculate the volumes of solids. It introduces the concept of slicing solids into thin pieces, approximating the volume of each slice, adding up the approximations, and taking the limit to get a definite integral. This process of "slice, approximate, integrate" can be used to find volumes of solids of revolution generated by revolving plane regions about axes, whether the slices are disks, washers, or other cross-sectional shapes. Examples are provided of finding volumes of solids rotated about axes using this method with disks, washers, and other cross-sectional areas.
1) A Riemann sum is the approximation of the area under a curve obtained by adding rectangles of various widths.
2) The definite integral of a function f from a to b, written as ∫abf(x)dx, is defined as the limit of Riemann sums as the widths of the subintervals approach 0.
3) The definite integral calculates the exact area under a curve between the bounds a and b and is equal to the antiderivative of f evaluated between those bounds.
The document discusses calculating the area of a 2D region R. It explains that the area can be found by taking a ruler and measuring the span of R from x=a to x=b. The cross-sectional length L(x) is defined at each x value. The region is divided into subintervals with arbitrary points xi selected in each. Rectangles with base Δx and height L(xi) approximate the area in each subinterval. The Riemann sum of these rectangles approximates the total area of R. The mathematical definition of the area is given as the definite integral of the cross-section function from a to b, according to the Fundamental Theorem of Calculus. An example problem finds the area
1. Double integrals allow us to calculate the volume of a solid bounded above by a surface z=f(x,y) and below by a plane region R. This is done by setting up a Riemann sum of the volumes of thin rectangular prisms.
2. The limit of the Riemann sum as the number of rectangles goes to infinity gives the double integral, written as ∫∫R f(x,y) dA.
3. Example problems demonstrate calculating the double integral to find the volume in different plane regions R, including those bounded by curves such as y=x2 and y=x.
Zeph notices that a duck's turd covers an area on the sand equivalent to the shaded region R in a graph. The region R is bounded by the functions f(x) = 1/x and g(x) = sin(x) between the points they intersect, x = 1.1141571 and x = 2.7726047. Zeph is asked to determine the area of R, the volume when R is revolved about the y-axis, and the volume when R is used as the base for a solid where each cross-section is an equilateral triangle.
The document discusses using integrals to calculate the area between two curves. It provides examples of finding areas bounded above and below by functions, including cases where the boundary curves intersect and cases where graphical methods are needed to find approximate intersection points. The key formula given is that the area between curves y=f(x) and y=g(x) from x=a to x=b is the integral from a to b of f(x)-g(x) dx. Examples are worked out demonstrating the application of this formula.
The Trapezoidal rule approximates the area under a curve by dividing it into trapezoids and calculating their individual areas. It works by taking the ordinates at evenly spaced intervals along the x-axis and using the formula: Area = (1/2) * (first ordinate + last ordinate + 2 * sum of middle ordinates) * width. This provides an estimate of the definite integral. The more trapezoids used, the more accurate the estimate. The estimate will be an overestimate if the gradient is increasing and an underestimate if decreasing over the interval.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document provides 15 important questions on vector calculus concepts including directional derivatives, unit normals, solenoidal and irrotational vectors, and verification problems for Gauss's divergence theorem, Green's theorem, and Stokes' theorem. Example problems include finding directional derivatives, unit normals, determining if a vector is solenoidal or irrotational, evaluating line integrals, and verifying the vector calculus theorems for different bounding shapes and regions.
The document discusses integration and indefinite integrals. It covers determining integrals by reversing differentiation, integrating algebraic expressions like constants, variables, and polynomials. It also discusses determining the constant of integration and using integration to find equations of curves from their gradients. Examples are provided to illustrate integrating functions and finding volumes generated by rotating an area about an axis.
The document describes how to calculate the volume of a solid object using Cavalieri's principle. It involves partitioning the solid into thin cross-sectional slices and approximating the volume of each slice as a cylinder with the slice's cross-sectional area and thickness. The total volume is then approximated as the sum of the cylindrical slice volumes. As the number of slices approaches infinity, this sum approaches the actual volume calculated as the integral of the cross-sectional area function over the solid's distance range.
This document discusses calculating the surface area of revolution by rotating a curve around an axis. It defines the surface area for simple shapes like cylinders and cones. For more complex surfaces, it approximates the curve as a polygon and calculates the surface area of each band formed by rotating line segments. The surface area is then defined as the limit of these approximations, which is equivalent to a definite integral of 2π times the curve's distance from the axis of revolution. Formulas are provided for rotating curves defined by y=f(x) or x=g(y), and examples are worked out applying these formulas.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
27. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
x
28. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
x
29. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
c x
A(c) is the area from 0 to c
30. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
c x x
A(c) is the area from 0 to c
A(x) is the area from 0 to x
31. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
32. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
33. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
34. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
35. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
Ac h Ac
h = width of rectangle
h
36. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
Ac h Ac
h = width of rectangle
h
As the width of the rectangle decreases, the estimate becomes more
accurate.
38. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
39. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
40. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
41. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
42. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
43. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
b
A f x dx
a
44. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
b
A f x dx
a
F b F a
45. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
b
A f x dx
a
F b F a
where F x is the primitive function of f x
46. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
x= 2
47. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
48. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
49. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
50. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
51. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3
ii x 2 1dx
2
52. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2
2 3
53. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2
2 3
1 33 3 1 2 3 2
3 3
54. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2
2 3
1 33 3 1 2 3 2
3 3
22
3