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Warm-up: 1)If a particle has a velocity function defined by, find its acceleration function. 2)If a particle has an acceleration function defined by, what.

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Presentation on theme: "Warm-up: 1)If a particle has a velocity function defined by, find its acceleration function. 2)If a particle has an acceleration function defined by, what."— Presentation transcript:

1 Warm-up: 1)If a particle has a velocity function defined by, find its acceleration function. 2)If a particle has an acceleration function defined by, what is its velocity function? Is there more than one possibility?

2 Integration Section 6.1 & 6.2 The Area Under a Curve / Indefinite Integrals

3 The Rectangle Method for Finding Areas Read over Example on pg. 351 and draw a conclusion about the Area under of curve as it relates to the number of rectangular sub-intervals.

4 The Rectangle Method for Finding Areas Read over Example on pg. 351 and draw a conclusion about the Area under of curve as it relates to the number of rectangular sub-intervals. When we become more comfortable with integration we will use integrals to more accurately find the area under a curve.

5 Anti-differentiation (Integration) The opposite of derivatives (anti-derivatives) Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?

6 Anti-differentiation (Integration) The opposite of derivatives (anti-derivatives) Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)? Let’s assume

7 Anti-differentiation (Integration) The opposite of derivatives (anti-derivatives) Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)? Let’s assume

8 Anti-differentiation (Integration) The opposite of derivatives (anti-derivatives) Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)? Let’s assume Could work?

9 Anti-differentiation (Integration) The opposite of derivatives (anti-derivatives) Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)? Let’s assume Could work? How about ?

10 Indefinite Integrals The process of finding anti-derivatives is called Anti-Differentiation or Integration.

11 Indefinite Integrals The process of finding anti-derivatives is called Anti-Differentiation or Integration.

12 Indefinite Integrals The process of finding anti-derivatives is called Anti-Differentiation or Integration. can be written as using Integral Notation,

13 Indefinite Integrals The process of finding anti-derivatives is called Anti-Differentiation or Integration. can be written as using Integral Notation, where the expression is called an Indefinite Integral,

14 Indefinite Integrals The process of finding anti-derivatives is called Anti-Differentiation or Integration. can be written as using Integral Notation, where the expression is called an Indefinite Integral, the function f(x) is called the Integrand,

15 Indefinite Integrals The process of finding anti-derivatives is called Anti-Differentiation or Integration. can be written as using Integral Notation, where the expression is called an Indefinite Integral, the function f(x) is called the Integrand, and the constant C is called the Constant of Integration.

16 Properties of Integrals:

17 A constant Factor can be moved through an Integral sign:

18 Properties of Integrals: A constant Factor can be moved through an Integral sign:

19 Properties of Integrals: A constant Factor can be moved through an Integral sign: An anti-derivative of a sum is the sum of the anti-derivatives:

20 Properties of Integrals: A constant Factor can be moved through an Integral sign: An anti-derivative of a sum is the sum of the anti-derivatives:

21 Properties of Integrals: A constant Factor can be moved through an Integral sign: An anti-derivative of a sum is the sum of the anti-derivatives: An anti-derivative of a difference is the difference of the anti-derivatives:

22 Properties of Integrals: A constant Factor can be moved through an Integral sign: An anti-derivative of a sum is the sum of the anti-derivatives: An anti-derivative of a difference is the difference of the anti-derivatives:

23 Integral Power Rule To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.

24 Integral Power Rule To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent. Find

25 Integral Power Rule To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent. Find

26 Integral Power Rule To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent. Find

27 Integral Power Rule To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent. Find

28 Examples (S) 1) Find 2) Find 3) Find

29 Examples (S) 1) Find 2) Find 3) Find

30 Examples 1) Find 2) Find 3) Find

31 Examples 1) Find 2) Find 3) Find

32 Examples 1) Find 2) Find 3) Find

33 Examples 1) Find 2) Find 3) Find

34 Examples 1) Find 2) Find 3) Find

35 Examples 1) Find 2) Find 3) Find

36 Examples of Common Integrals 1)Find 2)Find

37 Examples of Common Integrals 1)Find 2)Find

38 Examples of Common Integrals 1)Find 2)Find

39 Integral Formulas to Memorize The same as all of the derivative formulas that are memorized. List on pg. 357 (and inside front cover of textbook).

40 More Difficult Examples 1)Find 2)Find

41 More Difficult Examples 1)Find 2)Find

42 More Difficult Examples 1)Find 2)Find

43 More Difficult Examples 1)Find 2)Find

44 More Difficult Examples 1)Find 2)Find

45 More Examples (S) 3)Find 4)Find

46 More Examples (S) 3)Find 4)Find

47 More Examples 3)Find 4)Find

48 More Examples 3)Find 4)Find

49 More Examples 3)Find 4)Find

50 Last Example 5)Find

51 Last Example 5)Find

52 Last Example 5)Find

53 Last Example 5)Find

54 Homework: page 363 # 9 – 33 odd

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