Chapter 6 | Applications of Integration
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other. • The geometry of the functions and the difficulty of the integration are the main factors in deciding which integration method to use. 6.4 Arc Length of a Curve and Surface Area • The arc length of a curve can be calculated using a definite integral. • The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of y . • The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution. • The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals. 6.5 Physical Applications • Several physical applications of the definite integral are common in engineering and physics. • Definite integrals can be used to determine the mass of an object if its density function is known. • Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. • Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid. 6.6 Moments and Centers of Mass • Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system. • For point masses distributed along a number line, the moment of the system with respect to the origin is M = ∑ i =1 n m i x i . For point masses distributed in a plane, the moments of the system with respect to the x - and y -axes, respectively, are M x = ∑ i =1 n m i y i and M y = ∑ i =1 n m i x i , respectively. • For a lamina bounded above by a function f ( x ), the moments of the system with respect to the x - and y -axes, respectively, are M x = ρ ∫ a b ⎡ ⎣ f ( x ) ⎤ ⎦ 2 2 dx and M y = ρ ∫ a b xf ( x ) dx . • The x - and y -coordinates of the center of mass can be found by dividing the moments around the y -axis and around the x -axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line. • The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region. 6.7 Integrals, Exponential Functions, and Logarithms • The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way. • The cornerstone of the development is the definition of the natural logarithm in terms of an integral. • The function e x is then defined as the inverse of the natural logarithm. • General exponential functions are defined in terms of e x , and the corresponding inverse functions are general logarithms.
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