5.3/5.4 Definite Integrals Tues Feb 23

1
5.3/5.4 Definite IntegralsTues Feb 23

• Do Now
• Evaluate the following
• 1)
• 2)

2
HW Review worksheet p.349 3-6 35-38

• 3) 1.328125
• 4) 4.625
• 5) 2.7265625
• 6) 17.75
• 35) L 1.81 R 1.67
• 36) L .87 R .63
• 37) L 3.8 R 4.32
• 38) L 3.08 R 3.08

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More and More Rectangles

• When approximating area, the more rectangles we
  use, the space is covered under the curve
• More rectangles better approximation

4
Area Under a Curve

• For a function f defined on the interval a,b,
  if f is continuous on a,b and f(x) gt 0 on
  a,b, the area A under the curve y f(x) on
  a,b is given by

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Definite Integral

• The definite integral of f on the interval a,b
  is
• for any function on the interval a, b

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Fundamental Theorem of Calculus

• If f is continuous on a,b and F(x) is the
  antiderivative of f, then
• Step 1 Integrate f(x)
• Step 2 Plug in b and a into x
• Step 3 Subtract the 2 values

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Notation

• Note
• The left side is often used instead of the right
  side

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Area above and below x-axis

• Any area can be bounded by the x-axis and the
  function
• If the area is above the x-axis, then it is
  considered positive
• If the area is below the x-axis, then it is
  considered negative

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Reversing the Limits of Integration

• If we reverse the limits of integration,

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Additivity for Adjacent Intervals

• Let , and assume that f(x) is integrable.
• Then
• This is useful for absolute value or piecewise
  functions

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Examples

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You try

• 1)
• 2)
• 3)
• 4)
• 5)

13
Closure

• Hand in Compute the integral
• HW p.314 5-37 odds

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5.3 Definite Integral PracticeWed Feb 24

• Do Now
• Evaluate each integral
• 1)
• 2)

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HW p.314 5-37 odds

• 5) 27/2 23)
• 7) -1 25)
• 9) 128 27) 3/2
• 11) 27/2 29)
• 13) 16/3 31)
• 15) 31/40 33) e - 1
• 17) 2/3 35)
• 19) 12 37) ln 5
• 21) 11/6

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Remember!

• When using trig functions and your calculator,
  you must use radians!

17
Practice

• Worksheet p.372 6-11 14-22 27

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Closure

• Hand in Evaluate the integral
• HW Worksheet p.372 6-11, 14-22 27

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5.4 Fundamental Thm of Calculus Pt 2Thurs Feb 25

• Do Now
• Evaluate the integral

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HW Review p.372 6-11

• 6) 3
• 7) 0
• 8) 8
• 9) 88/3
• 10) 5
• 11) 12/5

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HW Review p.372 14-22, 27

• 14) 0 21)
• 15) 22)
• 16) 1 27) 16/3
• 17) 3
• 18) 0
• 19)
• 20)

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Fundamental Thm Part 2

• If f is continuous on a,b and
• then on a,b

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In words,

• If we differentiate an integral, they will cancel
  out.
• If the upper bound is a variable x, we can
  replace the current variable with x.
• NOTE This is extremely helpful when we dont
  know how to integrate something!

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Fundamental Thm Part 2

• If the upper bound is something more than x, then
  we also need to multiply our answer by the
  derivative

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Examples

• Ex 5.7 and 5.8
• Find F(x)

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You try Find F(x)

• 1)
• 2)
• 3)

27
Closure

• Hand in Find F(x) if
• HW p.320 7-15 odds 21 23 29 31 33
• Quiz Tues March 1

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5.4 Review/PracticeFri Feb 26

• Do Now
• Calculate the derivatives
• 1)
• 2)

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HW Review p.320 7-15 21 23 29-33

• 7) 29)
• 9) 1 cos x 31)
• 11) 33)
• 13)
• 15)
• 21)
• 23) sec(5t 9)

30
5.1-5.4 Quiz Review

• 5.1 Approximation of Area under Curve
• Left, Right, Midpoint Approximations
• 5.3/4 Fundamental Theorem of Calculus
• Definite Integrals
• Derivatives of Integrals

31
Practice

• 5.3 (all) and 5.4 (skip 3,5) Worksheet

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Closure

• What are the 2 parts of the Fundamental Theorem
  of Calculus? What can they be used for?
• Finish 5.3 5.4 worksheet
• 5.1-5.4 Tues March 1

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5.1-5.4 ReviewMon Feb 29

• Do Now
• Evaluate each integral
• 1)
• 2) Find the area under f(x) x2 on the interval
  0, 2 using 4 right endpoint rectangles

34
HW Review worksheet

• 5.3 5.4
• 1) 2 1a) 0
• 2a) 0 1b) tan(pi/4) 1
• 2b) 16/3 2a)
• 2c) 20/9 ln 9/4 2b)
• 2d) sqrt 2 4a)
• 2e) 28/3 4b)

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5.1-5.4 Quiz Review

• Approximation of Area under Curve
• Left, Right, Midpoint Approximations
• Fundamental Theorem of Calculus
• Definite Integrals
• Derivatives of Integrals
• All rules from 4.9 (Antiderivatives) will be
  needed!

36
Last call for questions

• 4.9 Antiderivatives
• 5.1 Rectangle Approximations
• 5.2/5.3 Definite Integrals
• 5.4 Derivatives of Integrals

37
Closure

• Make up a possible quiz problem that you are
  confident you can solveand solve it.
• Quiz Tomorrow