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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 is its velocity function? Is there more than one possibility?

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

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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.

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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.

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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)?

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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

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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

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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?

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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 ?

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

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

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

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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,

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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,

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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.

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Properties of Integrals:

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A constant Factor can be moved through an Integral sign:

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

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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:

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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:

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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:

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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:

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

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

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

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

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

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

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

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Examples 1) Find 2) Find 3) Find

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Examples 1) Find 2) Find 3) Find

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Examples 1) Find 2) Find 3) Find

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Examples 1) Find 2) Find 3) Find

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Examples 1) Find 2) Find 3) Find

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Examples 1) Find 2) Find 3) Find

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Examples of Common Integrals 1)Find 2)Find

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Examples of Common Integrals 1)Find 2)Find

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Examples of Common Integrals 1)Find 2)Find

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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).

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More Difficult Examples 1)Find 2)Find

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More Difficult Examples 1)Find 2)Find

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More Difficult Examples 1)Find 2)Find

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More Difficult Examples 1)Find 2)Find

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More Difficult Examples 1)Find 2)Find

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More Examples (S) 3)Find 4)Find

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More Examples (S) 3)Find 4)Find

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More Examples 3)Find 4)Find

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More Examples 3)Find 4)Find

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More Examples 3)Find 4)Find

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Last Example 5)Find

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Last Example 5)Find

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Last Example 5)Find

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Last Example 5)Find

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Homework: page 363 # 9 – 33 odd

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