Chapter 5: The Integral
Differential Equations
Equations that contain derivatives of unknown functions
Average Value Theorem
Every continuous function on a closed interval attains its average value at some point on the interval.
Antiderivative
F'(x)=f(x)
inscribed circumscribed
Given a plane region R whose area they sought, they worked both with a polygon P _________ in R and with a polygon Q _______ about R.
The Most General Antiderivative
If F'(x) = f(x) at each point of the open interval I, then every antiderivative G of f on I has the form G(x) = F(x) + C, where C is a constant
Evaluation of Integrals
If G is an antiderivative of the continous function f on the interval [a,b], then ∫ab f(x) dx = G(b) - G(a)
Existence of the Integral
If the function f is continuous on [a,b], then f is integrable on [a,b].
∫ac f(x) dx + ∫cb f(x) dx
Interval Union Property If a<c<b, then ∫ab f(x) dx = ?
The Area Between Two Curves
Let f and g be continuous with f(x)≥g(x) fo x in [a,b]. Then the area A of the region bounded by the curves y=f(x) and y = g(x) and by the vertical lines x = a and x= b is A= ∫ab [f(x) - g(x)] dx
Left-Endpoint Approximation
Ln=(∆x)(y₀+y₁+y₂+. . .+yn-₁)
partition
P of [a,b] is a collection of subintervals [x₀,x₁], [x₁,x₂], [x₂,x₃], . . . [xn-1, xn]
Right-Endpoint Approximation
Rn=(∆x)(y₁+y₂+y₃. . .+yn)
Fundamental Theorem of Calculus
Suppose that f is continuous on the closed interval [a,b] 1. If the function F is defined on [a,b] by F(x)=∫ax f(t) dt then F is an antiderivative of f. That is F'(x) = f(x) for x in [a,b] 2. If G is any antiderivative of f on [a,b], then ∫ab f(x) dx = [G(x)]ab = G(b) - G(a)
Integration by Substitution
Suppose that the function f has a continuous derivative on [a,b] and that f is continuous on the set g([a,b]). Let u=g(x). Then ∫ab f(g(x)) × g'(x) dx = ∫g(a)g(b) f(u) du.
signed minus
The Riemann sum R is then the sum of the _______ ares of these rectangles--that it, the sum of the areas of those rectangles that lie above the x-axis _______ the sum of the areas of those that lie below the x-axis.
indefinite integral
The collection of all antiderivatives of the function f(x) is called the ________ ________.
height function
The derivative of the area function A(x) is the curve's _______ _______ f(x)
The Integral as a Limit of a Sequence
The function f is integrable on [a,b] with integral I if and only if lim(x→∞)(Rn) = I for every sequence of Riemann sums associated with a sequence of partitions of [a,b] such that |Pn|→0 as n→+∞
acceleration
The particles _______ a(t) is the time derivative of its velocity.
velocity
The particles _______ v(t) is the time derivative of its position function.
Antidifferentiation
The process of finding a function from its derivatives, which is the opposite of differentiation.
Riemann sum
The sum is calculated by dividing the region up into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together.
x
What is the antiderivative of 1?
x²
What is the antiderivative of 2x?
sin x
What is the antiderivative of cos x?
-½cos 2x
What is the antiderivative of sin 2x?
¼x⁴
What is the antiderivative of x³?
c ∫ab f(x) dx
What is the integral of a constant multiple property? ∫ab cf(x) dx Thus a constant can be "moved across" the integral sign.
c(b-a)
What is the integral of a constant? ∫ab c dx This property is intuitively obvious because the area represented by the integral is simply a rectangle with base b-a and height c.
∫ab f(x) dx + ∫ab g(x) dx
What is the integral of a sum? ∫ab [f(x) + g(x)] dx?
Average Value
y(hat) = 1/b-a ∫ab f(x) dx
The integral of f from a to be
∫ab f(x) dx How would you say this?
Elementary Function
A function that can be expressed in terms of polynomial, trigonometric, exponential, or logarithmic functions by means of finite combinations of sums differences, products, quotients, roots and function composition.
many one
A single function has _____ antiderivatives, whereas a single function can only have ____ derivative
motion
Antidifferentiation enables us, in many important cases, to analyze the ________ of a particle (or "mass point") in terms of the forces acting on it.
position function
Antidifferentiation enables us, in many important cases, to analyze the motion of a particle (or "mass point") in terms of the forces acting on it. If the particle moves in rectilinear motion along a straight line--the x-axis for instance--under the influence of a given (possibly variable) force, then the motion of the particle is described by its _________ _______ x=x(t)
Integration by Substitution
Can be derived from the fundamental theorem of calculus. The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be used from left to right or from right to left in order to simplify a given integral.