Review: 4'8 Antiderivatives

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Review 4.8 Antiderivatives

• Finding Antiderivatives

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Antiderivative Formulas

• 2. Antiderivative Formulas
• f(x)xn,
• F(x) xn1C
• f(x)sin(kx)
• F(x)- cos(kx)C
• f(x)cos(kx)
• F(x) sin(kx)C
• f(x)sec2x
• F(x)tanxC

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Antiderivative Formulas

• 5) f(x)csc2x
• F(x)-cot xC
• f(x)sec x tan x
• F(x)sec x C
• f(x)csc x cotx
• F(x) -csc x C

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Antidifferentiation Rules

• 3. Antiderivative Linearity Rules
• function general antiderivative
• 1) kf(x), kF(x)C
• 2) f(x)g(x) F(x)G(x)C
• 3) f(x)-g(x) F(x)-G(x)C

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Initial Value Problems and Differential Equations

• 4. Differential Equatons
• To solve an initial value problem
• Find the general solutions for antiderivatives
• Use the initial value condition to determine the
  constant C.

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Indefinite Integrals

• 5. Indefinite Integrals
• 1) Defintion
• The collection of all antiderivatives of f is
  called the indefinite integral of f, and is
  denoted by
• The function f is called the integrand of the
  integral.
• It is a special symbol to represent the general
  antiderivative.

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Examples

• 2) Examples
• a)
• b)
• c)

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Discussion of Homework Problems

• page 314-316.

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Chapter 5 Integration

• Estimating with finite sums
• Sigma notation and limits of finite sums
• The definite integral
• The fundamental Theorem of Calculus
• Indefinite integral and substitution rule
• Substitution and area between two curves

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5.1 Estimating with Finite Sums

• Area
• Ex. Estimate the area of the region R under the
  graph of f(x)x21 from x0 to x2.
• Solution Let A be the area of the region R. We
  use two different rectangles
• inscribed rectangles and circumscribed
  rectangles.
• Divide the interval 0,2 into two equal parts.

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• Now we have two intervals0,1 and 1,2.
• First, we use the circumscribed rectangles.
• The total area of the circumscribed rectangles
  is
• 1(2)1(5)7 --an upper sum
• The sum of areas of circumscribed rectangles is
  called an upper sum.
• The total area of the inscribed rectangles is
• (1)(1)(1)(2)3 -- a lower sum
• The sum of areas of circumscribed rectangles is
  called a lower sum.

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• So 3ltAlt7.
• Now let us divide the interval 0,2 into four
  equal parts. 0, ½, 1/2, 1, 1, 3/2, 3/2,2
• So the upper sum
• 1/2(5/4)1/2(2)1/2(13/4)1/2(5)
• 5/8113/85/2
• 23/4
• the lower sum
• 1/2(1)1/2(5/4)1/2(2)1/2(13/4)
• 1/25/8113/8
• 15/4
• So 15/4ltAlt23/4.

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Summary

• By taking more and more rectangles, with each
  rectangle thinner than before, the finite sums
  give a better and better approximation to the
  true area of the region R.

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Practice

• 2 on page 333.

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5.2 Sigma Notation and Limits of Finite Sums

• Finite Sums and Sigma Notation
• Sigma Notation
• The sum of n terms is written as
• The Greek letter stands for sum. The index
  of summation k tells where the sum begins and
  where the sum ends

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Examples

• 2) Examples
• (a)
• (b)

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Algebra Rules

• 3) Algebra Rules
• Sum rule
• Difference Rule
• Constant Multiple Rule
• Constant Value Rule

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Some Formulas

• a)
• b)
• c)

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Examples

• a)
• b)

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Limits of Finite Sum

• 2. Limits of Finite Sum
• By taking more and more rectangles, with each
  rectangle thinner than before, the finite sums
  (both lower sum and upper sum) give a better and
  better approximation to the true area of the
  region R. So the limit of the finite sum is the
  true area.

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Riemann Sums

• 4. Riemann Sums
• We begin with any function f defined on
• a, b. Now divide the interval a, b into n
  subintervals.
• a, x1, x1, x2, x2, x3, ., xn-1, b
• The set Pa, x1, x2,.., b
• x0, x1, x2,.., xn
• is called a partition of a, b

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• In each subinterval xk-1, xk we select a point
  ck. Then the area of the small rectangle over
  xk-1, xk with height f(ck) is
• f(ck)(xk- xk-1) f(ck) ? xk
• We sum all these areas to get
• This sum is called a Riemann sum for f on the
  interval a, b. The largest width of all
  subintervals is called the norm of the partition
  P and is denoted by P.

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Practice

• 2, 8, 12, 18, 20 on page 342-343.

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Homework

• 1,3, 5 on page 333
• 1-11 odd, 15, 17, 19, 23 on page 342-343.