Calculus Volume 2

Key Terms

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 $y=acosh(x/a)y=acosh(x/a)$ 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
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 $(ln2)/k(ln2)/k$
exponential decay
systems that exhibit exponential decay follow a model of the form $y=y0e−kty=y0e−kt$
exponential growth
systems that exhibit exponential growth follow a model of the form $y=y0ekty=y0ekt$
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 $(ln2)/k(ln2)/k$
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, $F=kx,F=kx,$ where $kk$ 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 $M=∑i=1nmixi;M=∑i=1nmixi;$ if, instead, we consider a region in the plane, bounded above by a function $f(x)f(x)$ over an interval $[a,b],[a,b],$ then the moments of the region with respect to the x- and y-axes are given by $Mx=ρ∫ab[f(x)]22dxMx=ρ∫ab[f(x)]22dx$ and $My=ρ∫abxf(x)dx,My=ρ∫abxf(x)dx,$ 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
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