 # 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 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:

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?

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

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.

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.

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)?

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

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

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?

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 ?

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

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

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

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,

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,

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.

Properties of Integrals:

A constant Factor can be moved through an Integral sign:

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

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:

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:

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:

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:

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

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

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

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

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

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

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

Examples 1) Find 2) Find 3) Find

Examples 1) Find 2) Find 3) Find

Examples 1) Find 2) Find 3) Find

Examples 1) Find 2) Find 3) Find

Examples 1) Find 2) Find 3) Find

Examples 1) Find 2) Find 3) Find

Examples of Common Integrals 1)Find 2)Find

Examples of Common Integrals 1)Find 2)Find

Examples of Common Integrals 1)Find 2)Find

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).

More Difficult Examples 1)Find 2)Find

More Difficult Examples 1)Find 2)Find

More Difficult Examples 1)Find 2)Find

More Difficult Examples 1)Find 2)Find

More Difficult Examples 1)Find 2)Find

More Examples (S) 3)Find 4)Find

More Examples (S) 3)Find 4)Find

More Examples 3)Find 4)Find

More Examples 3)Find 4)Find

More Examples 3)Find 4)Find

Last Example 5)Find

Last Example 5)Find

Last Example 5)Find

Last Example 5)Find

Homework: page 363 # 9 – 33 odd

Download ppt "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."

Similar presentations