arc length: the arc length of a curve can be thought of as the distance a person would travel along the path of the curve

catenary: 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

center of mass: the point at which the total mass of the system could be concentrated without changing the moment

centroid: 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

density function: a density function describes how mass is distributed throughout an object; it can be a linear density, 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

disk method: a special case of the slicing method used with solids of revolution when the slices are disks

doubling time: if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by $(ln 2) / k$

exponential decay: systems that exhibit exponential decay follow a model of the form $y = y_{0} e^{− k t}$

exponential growth: systems that exhibit exponential growth follow a model of the form $y = y_{0} e^{k t}$

frustum: a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base

half-life: 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 $(ln 2) / k$

Hooke’s law: 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, $F = k x,$ where $k$ is a constant

lamina: a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional

method of cylindrical shells: 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

moment: if n masses are arranged on a number line, the moment of the system with respect to the origin is given by $M = \sum_{i = 1}^{n} m_{i} x_{i};$ if, instead, we consider a region in the plane, bounded above by a function $f (x)$ over an interval $[a, b],$ then the moments of the region with respect to the x- and y-axes are given by $M_{x} = ρ \int_{a}^{b} \frac{{[f (x)]}^{2}}{2} d x$ and $M_{y} = ρ \int_{a}^{b} x f (x) d x,$ respectively

slicing method: 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

solid of revolution: a solid generated by revolving a region in a plane around a line in that plane

surface area: the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces

symmetry principle: the symmetry principle states that if a region R is symmetric about a line l, then the centroid of R lies on l

theorem of Pappus for volume: this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis is equal to the area of the region multiplied by the distance traveled by the centroid of the region

washer method: a special case of the slicing method used with solids of revolution when the slices are washers

work: the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance