Chapter 1

Integration

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Summary

Riemann sums allow for much flexibility in choosing the set of points { x i * } at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum Left- and right-endpoint approximations are special kinds of Riemann sums where the values of { x i * } are chosen to be the left or right endpoints of the subintervals, respectively The definite integral can be used to calculate net signed area, which is the area above the x -axis less the area below the x -axis. The width of each rectangle is Δ x = b - a n

Key terms

definite integral

a primary operation of calculus; the area between the curve and the x -axis over a given interval is a definite integral

net signed area

the area between a function and the x -axis such that the area below the x -axis is subtracted from the area above the x -axis; the result is the same as the definite integral of…

right-endpoint approximation

the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of…

riemann sum

an estimate of the area under the curve of the form A ≈ ∑ i = 1 n f ( x i * ) Δ x

lower sum

a sum obtained by using the minimum value of f ( x ) on each subinterval

upper sum

a sum obtained by using the maximum value of f ( x ) on each subinterval

integrable function

a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as n goes to infinity exists

Chapter 2

Applications of Integration

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Summary

Using the slicing method, we can find a volume by integrating the cross-sectional area If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals and add the results to find the area of the region The principles are the same regardless of which variable is used as the variable of integration

Key terms

principles

the same regardless of which variable is used as the variable of integration

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…

region

complex, use the absolute value of the difference of the functions

cross-section

the intersection of a plane and a solid object

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

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…

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…

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

Chapter 3

Techniques of Integration

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Summary

Integration by parts applies to both definite and indefinite integrals To integrate products involving sin ( a x ) , sin ( b x ) , cos ( a x ) , and cos ( b x ) , use the substitutions The integration-by-parts formula allows the exchange of one integral for another, possibly easier, integral Integrals of trigonometric functions can be evaluated by the use of various strategies.

Key terms

integration by parts

a technique of integration that allows the exchange of one integral for another using the formula ∫ ​ u d v = u v - ∫ ​ v d u

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…

midpoint rule

a rule that uses a Riemann sum of the form M n = ∑ i = 1 n f ( m i ) Δ x , where m i is the midpoint of the i th subinterval to approximate ∫ a b f ( x ) d x

Simpson’s rule

a rule that approximates ∫ a b f ( x ) d x using the integrals of a piecewise quadratic function. The approximation S n to ∫ a b f ( x ) d x is given by S n = Δ x 3 ( f ( x 0 ) +…

absolute error

if B is an estimate of some quantity having an actual value of A , then the absolute error is given by | A - B |

numerical integration

the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson’s rule

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

trigonometric substitution

an integration technique that converts an algebraic integral containing expressions of the form a 2 - x 2 , a 2 + x 2 , or x 2 - a 2 into a trigonometric integral

Chapter 4

Introduction to Differential Equations

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Summary

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. A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation Euler’s Method is a numerical technique that can be used to approximate solutions to a differential equation

Key terms

differential equation

an equation involving a function y = y ( x ) and one or more of its derivatives

solution

a function y = f ( x ) that satisfies the differential equation when f and its derivatives are substituted into the equation

direction field

a mathematical object used to graphically represent solutions to a first-order differential equation

order of a differential equation

the highest order of any derivative of the unknown function that appears in the equation

Euler’s Method

a numerical technique used to approximate solutions to an initial-value problem

initial-value problem

a differential equation together with an initial value or values

initial value

called an initial-value problem

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…

Chapter 5

Sequences and Series

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Summary

Given the infinite series ∑ n = 1 ∞ a n = a 1 + a 2 + a 3 + ⋯ ∑ n = 1 ∞ a n = a 1 + a 2 + a 3 + ⋯ and the corresponding sequence of partial sums { S k } where S k = ∑ n = 1 k a n = a 1 + a 2 + a 3 + ⋯ + a k , S k = ∑ n… To determine the convergence of a sequence given by an explicit formula a n = f ( n ) , we use the properties of limits for functions If { a n } and { b n } are convergent sequences that converge to A and B , respectively, and c is any real number, then the sequence { c a n } converges to c · A , the sequences { a n ± b n } converge to A ± B , the… If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone

Key terms

sequence

an ordered list of numbers of the form a 1 , a 2 , a 3 ,… is a sequence

convergent sequence

a convergent sequence is a sequence { a n } for which there exists a real number L such that a n is arbitrarily close to L as long as n is sufficiently large

geometric sequence

a sequence { a n } in which the ratio a n + 1 / a n is the same for all positive integers n is called a geometric sequence

geometric series

a geometric series is a series that can be written in the form ∑ n = 1 ∞ a r n - 1 = a + a r 2 + a r 3 + ⋯ ∑ n = 1 ∞ a r n - 1 = a + a r 2 + a r 3 + ⋯

partial sum

the k th partial sum of the infinite series ∑ n = 1 ∞ a n is the finite sum S k = ∑ n = 1 k a n = a 1 + a 2 + a 3 + ⋯ + a k S k = ∑ n = 1 k a n = a 1 + a 2 + a 3 + ⋯ + a k

infinite series

an infinite series is an expression of the form a 1 + a 2 + a 3 + ⋯ = ∑ n = 1 ∞ a n a 1 + a 2 + a 3 + ⋯ = ∑ n = 1 ∞ a n

term

the number a n in the sequence { a n } is called the n th term of the sequence

divergent sequence

a sequence that is not convergent is divergent

Chapter 6

Power Series

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Summary

For a power series centered at x = a , one of the following three properties hold: The power series converges only at x = a . In this case, we say that the radius of convergence is R = 0 The power series converges for all real numbers x . In this case, we say that the radius of convergence is R = ∞

Key terms

power series

a series of the form ∑ n = 0 ∞ c n x n is a power series centered at x = 0 ; a series of the form ∑ n = 0 ∞ c n ( x - a ) n is a power series centered at x = a

radius of convergence

if there exists a real number R > 0 such that a power series centered at x = a converges for | x - a | < R and diverges for | x - a | > R , then R is the radius of convergence…

binomial series

the Maclaurin series for f ( x ) = ( 1 + x ) r ; it is given by ( 1 + x ) r = ∑ n = 0 ∞ ( r n ) x n = 1 + r x + r ( r - 1 ) 2 ! x 2 + ⋯ + r ( r - 1 ) ⋯ ( r - n + 1 ) n ! x n + ⋯…

Taylor polynomials

the n th Taylor polynomial for f at x = a is p n ( x ) = f ( a ) + f ′ ( a ) ( x - a ) + f ″ ( a ) 2 ! ( x - a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x - a ) n p n ( x ) = f ( a ) + f ′ (…

Maclaurin polynomial

a Taylor polynomial centered at 0; the n th Taylor polynomial for f at 0 is the n th Maclaurin polynomial for f

Maclaurin series

a Taylor series for a function f at x = 0 is known as a Maclaurin series for f

Chapter 7

Parametric Equations and Polar Coordinates

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Summary

Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve

Key terms

parametric equations

the equations x = x ( t ) and y = y ( t ) that define a parametric curve

angular coordinate

θ the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial ( x ) axis, measured counterclockwise

parameter

an independent variable that both x and y depend on in a parametric curve; usually represented by the variable t

parametric curve

the graph of the parametric equations x ( t ) and y ( t ) over an interval a ≤ t ≤ b combined with the equations

cardioid

a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is r = a ( 1 + sin θ ) or r = a…

directrix

a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two

discriminant

the value 4 A C - B 2 , which is used to identify a conic when the equation contains a term involving x y , is called a discriminant

eccentricity

the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix