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5.3/5.4 Definite Integrals Tues Feb 23

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5.3/5.4 Definite Integrals Thurs Feb 19 Do Now Evaluate the following 1) 2) 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 ... – PowerPoint PPT presentation

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Title: 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

3
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

5
Definite Integral
  • The definite integral of f on the interval a,b
    is
  • for any function on the interval a, b

6
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

7
Notation
  • Note
  • The left side is often used instead of the right
    side

8
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

9
Reversing the Limits of Integration
  • If we reverse the limits of integration,

10
Additivity for Adjacent Intervals
  • Let , and assume that f(x) is integrable.
  • Then
  • This is useful for absolute value or piecewise
    functions

11
Examples

12
You try
  • 1)
  • 2)
  • 3)
  • 4)
  • 5)

13
Closure
  • Hand in Compute the integral
  • HW p.314 5-37 odds

14
5.3 Definite Integral PracticeWed Feb 24
  • Do Now
  • Evaluate each integral
  • 1)
  • 2)

15
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

16
Remember!
  • When using trig functions and your calculator,
    you must use radians!

17
Practice
  • Worksheet p.372 6-11 14-22 27

18
Closure
  • Hand in Evaluate the integral
  • HW Worksheet p.372 6-11, 14-22 27

19
5.4 Fundamental Thm of Calculus Pt 2Thurs Feb 25
  • Do Now
  • Evaluate the integral

20
HW Review p.372 6-11
  • 6) 3
  • 7) 0
  • 8) 8
  • 9) 88/3
  • 10) 5
  • 11) 12/5

21
HW Review p.372 14-22, 27
  • 14) 0 21)
  • 15) 22)
  • 16) 1 27) 16/3
  • 17) 3
  • 18) 0
  • 19)
  • 20)

22
Fundamental Thm Part 2
  • If f is continuous on a,b and
  • then on a,b

23
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!

24
Fundamental Thm Part 2
  • If the upper bound is something more than x, then
    we also need to multiply our answer by the
    derivative

25
Examples
  • Ex 5.7 and 5.8
  • Find F(x)

26
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

28
5.4 Review/PracticeFri Feb 26
  • Do Now
  • Calculate the derivatives
  • 1)
  • 2)

29
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

32
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

33
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)

35
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
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