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Introduction

  • Chapter 6 of Calculus typically covers the concepts of integration and accumulation of change.

  • Key topics include areas between curves, volumes by slicing, volumes of revolution, arc length, surface area, and physical applications.

  • The chapter also delves into moments and centers of mass, integrals involving exponential functions and logarithms, exponential growth and decay, and hyperbolic functions.

  • Understanding these concepts is crucial for solving problems related to areas, volumes, and other physical applications in engineering and physics.

Areas Between Curves [1]

  • Concept: Definite integrals can be used to find the area between two curves.

  • Method: Integrate the difference of the functions defining the curves.

  • Complex Regions: Use the absolute value of the difference if the graphs cross or the region is complex.

  • Alternative Approach: Sometimes it is easier to integrate with respect to y.

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Volumes by Slicing [2]

  • Concept: Definite integrals can be used to find the volumes of solids.

  • Method: Use the slicing method to find volume by integrating the cross-sectional area.

  • Disks: For solids of revolution, the volume slices are often disks with circular cross-sections.

  • Washers: If a solid has a cavity, use the method of washers by subtracting the inner circle's area from the outer circle's area before integrating.

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Volumes of Revolution [3]

  • Concept: The method of cylindrical shells is used to calculate the volume of a solid of revolution.

  • Preference: This method is sometimes preferable to disks or washers because it involves integrating with respect to the other variable.

  • Factors: The geometry of the functions and the difficulty of the integration determine the method used.

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Arc Length and Surface Area [4]

  • Concept: The arc length of a curve can be calculated using a definite integral.

  • Approximation: The arc length is first approximated using line segments, generating a Riemann sum.

  • Surface Area: The concepts used to calculate arc length can be generalized to find the surface area of a surface of revolution.

  • Difficulty: The integrals for arc length and surface area are often difficult to evaluate and may require a computer or calculator.

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Physical Applications [5]

  • Concept: Definite integrals have several physical applications in engineering and physics.

  • Mass: Integrals can determine the mass of an object if its density function is known.

  • Work: Work can be calculated from integrating a force function or counteracting gravity.

  • Force: Integrals can calculate the force exerted on an object submerged in a liquid.

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Moments and Centers of Mass [6]

  • Concept: The center of mass is the point where the total mass of a system could be concentrated without changing the moment.

  • Point Masses: For point masses along a number line, the moment is the sum of the product of mass and position.

  • Plane Masses: For point masses in a plane, moments are calculated with respect to the x- and y-axes.

  • Lamina: For a lamina bounded by a function, moments are calculated using integrals.

  • Centroid: The coordinates of the center of mass are found by dividing the moments by the total mass.

  • Symmetry: If a region is symmetric with respect to a line, the centroid lies on that line.

  • Pappus' Theorem: The volume of a solid formed by revolving a region around an axis is the area of the region multiplied by the distance traveled by the centroid.

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Exponential Functions and Logarithms [7]

  • Concept: Logarithms and exponential functions are defined rigorously in this section.

  • Natural Logarithm: Defined in terms of an integral.

  • Exponential Function: Defined as the inverse of the natural logarithm.

  • General Functions: General exponential functions and their inverses are defined.

  • Properties: Familiar properties of logarithms and exponents hold in this rigorous context.

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Exponential Growth and Decay [8]

  • Concept: Exponential growth and decay are common applications of exponential functions.

  • Growth Model: Systems that exhibit exponential growth follow the model y = y0ekt.

  • Growth Rate: The rate of growth is proportional to the quantity present.

  • Doubling Time: Systems with exponential growth have a constant doubling time given by (ln2)/k.

  • Decay Model: Systems that exhibit exponential decay follow the model y = y0e−kt.

  • Half-Life: Systems with exponential decay have a constant half-life given by (ln2)/k.

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Hyperbolic Functions [9]

  • Concept: Hyperbolic functions are defined in terms of exponential functions.

  • Differentiation: Term-by-term differentiation yields formulas for hyperbolic functions.

  • Integration: Differentiation formulas give rise to integration formulas.

  • Inverses: Hyperbolic functions have inverses with appropriate range restrictions.

  • Applications: Common physical applications include calculations involving catenaries.

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