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Chapter 6 | Applications of Integration
CHAPTER 6 REVIEW
KEY TERMS
arc length catenary the arc length of a curve can be thought of as the distance a person would travel along the path of the curve a curve in the shape of the function y = a cosh( x / a ) is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary the point at which the total mass of the system could be concentrated without changing the moment the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina corresponds to the centroid of the representative region the intersection of a plane and a solid object a density function describes how mass is distributed throughout an object; it can be a linear density, center of mass centroid cross-section density function expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume a special case of the slicing method used with solids of revolution when the slices are disks if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by (ln2)/ k systems that exhibit exponential decay follow a model of the form y = y 0 e − kt systems that exhibit exponential growth follow a model of the form y = y 0 e kt a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by (ln2)/ k this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, F = kx , where k is a constant the pressure exerted by water on a submerged object a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable if n masses are arranged on a number line, the moment of the system with respect to the origin is given by M = ∑ i =1 n m i x i ; if, instead, we consider a region in the plane, bounded above by a function f ( x ) over an interval disk method doubling time exponential decay exponential growth frustum half-life Hooke’s law hydrostatic pressure lamina method of cylindrical shells moment ⎡ ⎣ a , b ⎤ ⎦ , then the moments of the region with respect to the x - and y -axes are given by M x = ρ ∫ a b ⎡ ⎣ f ( x ) ⎤ ⎦ 2 2 dx and M y = ρ ∫ a b xf ( x ) dx , respectively a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume a solid generated by revolving a region in a plane around a line in that plane slicing method solid of revolution
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