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catenary : a curve in the shape of the functiony=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 | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
center of mass : the point at which the total mass of the system could be concentrated without changing the moment | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 correspond... | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
cross-section : the intersection of a plane and a solid object | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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... | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
disk method : a special case of the slicing method used with solids of revolution when the slices are disks | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
exponential decay : systems that exhibit exponential decay follow a model of the formy=y0eâkty=y0eâkt | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
exponential growth : systems that exhibit exponential growth follow a model of the formy=y0ekty=y0ekt | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
frustum : a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
Hookeâs 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,wherekkis a constant | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
hydrostatic pressure : the pressure exerted by water on a submerged object | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
lamina : a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
moment : ifnmasses are arranged on a number line, the moment of the system with respect to the origin is given byM=âi=1nmixi;M=âi=1nmixi;if, instead, we consider a region in the plane, bounded above by a functionf(x)f(x)over an interval[a,b],[a,b],then the moments of the region with respect to thex- andy-axes are g... | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 valu... | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
solid of revolution : a solid generated by revolving a region in a plane around a line in that plane | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
symmetry principle : the symmetry principle states that if a regionRis symmetric about a linel, then the centroid ofRlies onl | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
washer method : a special case of the slicing method used with solids of revolution when the slices are washers | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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 | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
The integration-by-parts formula allows the exchange of one integral for another, possibly easier, integral. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Integration by parts applies to both definite and indefinite integrals. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Integrals of trigonometric functions can be evaluated by the use of various strategies. These strategies includeApplying trigonometric identities to rewrite the integral so that it may be evaluated byu-substitutionUsing integration by partsApplying trigonometric identities to rewrite products of sines and cosines with ... | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Applying trigonometric identities to rewrite the integral so that it may be evaluated byu-substitution | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Using integration by parts | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Applying trigonometric identities to rewrite products of sines and cosines with different arguments as the sum of individual sine and cosine functions | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Applying reduction formulas | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
For integrals involvinga2âx2,a2âx2,use the substitutionx=asinθx=asinθanddx=acosθdθ.dx=acosθdθ. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
For integrals involvinga2+x2,a2+x2,use the substitutionx=atanθx=atanθanddx=asec2θdθ.dx=asec2θdθ. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
For integrals involvingx2âa2,x2âa2,substitutex=asecθx=asecθanddx=asecθtanθdθ.dx=asecθtanθdθ. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Partial fraction decomposition is a technique used to break down a rational function into a sum of simple rational functions that can be integrated using previously learned techniques. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
When applying partial fraction decomposition, we must make sure that the degree of the numerator is less than the degree of the denominator. If not, we need to perform long division before attempting partial fraction decomposition. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
The form the decomposition takes depends on the type of factors in the denominator. The types of factors include nonrepeated linear factors, repeated linear factors, nonrepeated irreducible quadratic factors, and repeated irreducible quadratic factors. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
An integration table may be used to evaluate indefinite integrals. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
A CAS (or computer algebra system) may be used to evaluate indefinite integrals. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
It may require some effort to reconcile equivalent solutions obtained using different methods. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
The most commonly used techniques for numerical integration are the midpoint rule, trapezoidal rule, and Simpsonâs rule. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Simpsonâs rule approximates the definite integral by first approximating the original function using piecewise quadratic functions. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Integrals of functions over infinite intervals are defined in terms of limits. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
Integrals of functions over an interval for which the function has a discontinuity at an endpoint may be defined in terms of limits. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
The convergence or divergence of an improper integral may be determined by comparing it with the value of an improper integral for which the convergence or divergence is known. | https://openstax.org/books/calculus-volume-2/pages/3-key-concepts |
â« u d v = u v â â« v d u â« u d v = u v â â« v d u | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
â« a b u d v = u v | a b â â« a b v d u â« a b u d v = u v | a b â â« a b v d u | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
sin ( a x ) sin ( b x ) = 1 2 cos ( ( a â b ) x ) â 1 2 cos ( ( a + b ) x ) sin ( a x ) sin ( b x ) = 1 2 cos ( ( a â b ) x ) â 1 2 cos ( ( a + b ) x ) | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
sin ( a x ) cos ( b x ) = 1 2 sin ( ( a â b ) x ) + 1 2 sin ( ( a + b ) x ) sin ( a x ) cos ( b x ) = 1 2 sin ( ( a â b ) x ) + 1 2 sin ( ( a + b ) x ) | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
cos ( a x ) cos ( b x ) = 1 2 cos ( ( a â b ) x ) + 1 2 cos ( ( a + b ) x ) cos ( a x ) cos ( b x ) = 1 2 cos ( ( a â b ) x ) + 1 2 cos ( ( a + b ) x ) | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
â« sec n x d x = sec n - 2 x tan x n â 1 + n â 2 n â 1 â« sec n â 2 x d x ; n â 1 â« sec n x d x = sec n - 2 x tan x n â 1 + n â 2 n â 1 â« sec n â 2 x d x ; n â 1 | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
â« tan n x d x = 1 n â 1 tan n â 1 x â â« tan n â 2 x d x â« tan n x d x = 1 n â 1 tan n â 1 x â â« tan n â 2 x d x | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
M n = â i = 1 n f ( m i ) Î x M n = â i = 1 n f ( m i ) Î x | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
T n = 1 2 Î x ( f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ⯠+ 2 f ( x n â 1 ) + f ( x n ) ) T n = 1 2 Î x ( f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ⯠+ 2 f ( x n â 1 ) + f ( x n ) ) | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
S n = Î x 3 ( f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) + 4 f ( x 5 ) + ⯠+ 2 f ( x n â 2 ) + 4 f ( x n â 1 ) + f ( x n ) ) S n = Î x 3 ( f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) + 4 f ( x 5 ) + ⯠+ 2 f ( x n â 2 ) + 4 f ( x n â 1 ) + f ( x n ) ) | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
â« a + â f ( x ) d x = lim t â + â â« a t f ( x ) d x â« â â b f ( x ) d x = lim t â â â â« t b f ( x ) d x â« â â + â f ( x ) d x = â« â â 0 f ( x ) d x + â« 0 + â f ( x ) d x â« a + â f ( x ) d x = lim t â + â â« a t f ( x ) d x â« â â b f ( x ) d x = lim t â â â ... | https://openstax.org/books/calculus-volume-2/pages/3-key-equations |
absolute error : ifBBis an estimate of some quantity having an actual value ofA,A,then the absolute error is given by|AâB||AâB| | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
computer algebra system (CAS) : technology used to perform many mathematical tasks, including integration | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
improper integral : an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
integration by parts : a technique of integration that allows the exchange of one integral for another using the formulaâ«âudv=uvââ«âvduâ«âudv=uvââ«âvdu | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
integration table : a table that lists integration formulas | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
midpoint rule : a rule that uses a Riemann sum of the formMn=âi=1nf(mi)Îx,Mn=âi=1nf(mi)Îx,wheremimiis the midpoint of theith subinterval to approximateâ«abf(x)dxâ«abf(x)dx | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
numerical integration : the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpsonâs rule | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
partial fraction decomposition : a technique used to break down a rational function into the sum of simple rational functions | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
power reduction formula : a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
relative error : error as a percentage of the absolute value, given by|AâBA|=|AâBA|·100%|AâBA|=|AâBA|·100% | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
Simpsonâs rule : a rule that approximatesâ«abf(x)dxâ«abf(x)dxusing the integrals of a piecewise quadratic function. The approximationSnSntoâ«abf(x)dxâ«abf(x)dxis given bySn=Îx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+â¯+2f(xnâ2)+4f(xnâ1)+f(xn))Sn=Îx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+â¯+2f(xnâ2)... | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
trigonometric integral : an integral involving powers and products of trigonometric functions | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
trigonometric substitution : an integration technique that converts an algebraic integral containing expressions of the forma2âx2,a2âx2,a2+x2,a2+x2,orx2âa2x2âa2into a trigonometric integral | https://openstax.org/books/calculus-volume-2/pages/3-key-terms |
A differential equation is an equation involving a functiony=f(x)y=f(x)and one or more of its derivatives. A solution is a functiony=f(x)y=f(x)that satisfies the differential equation whenffand its derivatives are substituted into the equation. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
Eulerâs Method is a numerical technique that can be used to approximate solutions to a differential equation. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
A separable differential equation is any equation that can be written in the formyâ²=f(x)g(y).yâ²=f(x)g(y). | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
The method of separation of variables is used to find the general solution to a separable differential equation. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
When studying population functions, different assumptionsâsuch as exponential growth, logistic growth, or threshold populationâlead to different rates of growth. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
Any first-order linear differential equation can be written in the formyâ²+p(x)y=q(x).yâ²+p(x)y=q(x). | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit. | https://openstax.org/books/calculus-volume-2/pages/4-key-concepts |
x n = x 0 + n h y n = y n â 1 + h f ( x n â 1 , y n â 1 ) , where h is the step size x n = x 0 + n h y n = y n â 1 + h f ( x n â 1 , y n â 1 ) , where h is the step size | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
y â² = f ( x ) g ( y ) y â² = f ( x ) g ( y ) | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
d u d t = INFLOW RATE â OUTFLOW RATE d u d t = INFLOW RATE â OUTFLOW RATE | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
d T d t = k ( T â T s ) d T d t = k ( T â T s ) | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
d P d t = r P ( 1 â P K ) , P ( 0 ) = P 0 d P d t = r P ( 1 â P K ) , P ( 0 ) = P 0 | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
Solution to the logistic differential equation/initial-value problem | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
P ( t ) = P 0 K e r t ( K â P 0 ) + P 0 e r t P ( t ) = P 0 K e r t ( K â P 0 ) + P 0 e r t | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
d P d t = â r P ( 1 â P K ) ( 1 â P T ) d P d t = â r P ( 1 â P K ) ( 1 â P T ) | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
y â² + p ( x ) y = q ( x ) y â² + p ( x ) y = q ( x ) | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
μ ( x ) = e ⫠p ( x ) d x μ ( x ) = e ⫠p ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/4-key-equations |
asymptotically semi-stable solution : y=ky=kif it is neither asymptotically stable nor asymptotically unstable | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
asymptotically stable solution : y=ky=kif there existsε>0ε>0such that for any valuecâ(kâε,k+ε)câ(kâε,k+ε)the solution to the initial-value problemyâ²=f(x,y),y(x0)=cyâ²=f(x,y),y(x0)=capproacheskkasxxapproaches infinity | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
asymptotically unstable solution : y=ky=kif there existsε>0ε>0such that for any valuecâ(kâε,k+ε)câ(kâε,k+ε)the solution to the initial-value problemyâ²=f(x,y),y(x0)=cyâ²=f(x,y),y(x0)=cnever approacheskkasxxapproaches infinity | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
autonomous differential equation : an equation in which the right-hand side is a function ofyyalone | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
carrying capacity : the maximum population of an organism that the environment can sustain indefinitely | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
differential equation : an equation involving a functiony=y(x)y=y(x)and one or more of its derivatives | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
direction field (slope field) : a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
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