### Key Terms

- 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 $y=a\phantom{\rule{0.2em}{0ex}}\text{cosh}\left(x\text{/}a\right)$ 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

- cross-section
- the intersection of a plane and a solid object

- 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 $\left(\text{ln}\phantom{\rule{0.2em}{0ex}}2\right)\text{/}k$

- exponential decay
- systems that exhibit exponential decay follow a model of the form $y={y}_{0}{e}^{\text{\u2212}kt}$

- exponential growth
- systems that exhibit exponential growth follow a model of the form $y={y}_{0}{e}^{kt}$

- 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 $\left(\text{ln}\phantom{\rule{0.2em}{0ex}}2\right)\text{/}k$

- 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, $F=kx,$ where $k$ is a constant

- hydrostatic pressure
- the pressure exerted by water on a submerged object

- 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 $M={\displaystyle \sum}_{i=1}^{n}{m}_{i}{x}_{i};$ if, instead, we consider a region in the plane, bounded above by a function $f(x)$ over an interval $\left[a,b\right],$ then the moments of the region with respect to the*x*- and*y*-axes are given by ${M}_{x}=\rho {\displaystyle {\int}_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx}$ and ${M}_{y}=\rho {\displaystyle {\int}_{a}^{b}xf(x)dx},$ 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