6.5: Average Value of a Function

Objectives:

• Interpret the Mean Value Theorem for Integrals geometrically.

• Find the average value of a continuous function.

Definitions and Formulas:

1. Average Value of a Function
2. The Mean Value Theorem for Integrals

Flashcards:

1. The Mean Value Theorem for Integrals

Examples:

An explanation of the Mean Value Theorem for Integrals

Ex1. Finding the average value of a function.

Ex2. Average velocity.

Student example


Problem Set:

# 1, 4, 7, 10, 13, 16, 22

6.3: Volumes by Cylindrical Shells

Objectives:

• Find the volume of a figure using the cylindrical shells method.

Read:
pgs. 433-436

Definitions/Formulas:
Definition of volume by cylindrical shells (formula)

Flashcards:
Definition of volume by cylindrical shells (formula)

Examples (will be posted 3/16/12):

Ex1. cylindrical shells


Ex2. more cylindrical shells


Ex3. even more cylindrical shells!


student example


Problem set 6.3
#2, 4, 5, 11, 20, 26, 33, 38, 43

6.2b: Volumes

Objectives:

• Use cross sections to find volumes of revolution (integrating with respect to y).

Read:
pg. 425 (example 3)
pg. 428 (example 6)

Definitions/ Formulas:
Definition of Volume

Flashcards:
Definition of Volume

Examples

Ex1. Volume of a solid of revolution (about the y-axis)

student example

Ex2. Volume of hollow objects of revolution (about the y-axis)

student example



Problem Set: #5, 9, 15, 21, 22, 24

6.2a: Volumes

Objectives:

• Use cross sections to find volumes of revolution (integrating with respect to x).
• Find volumes of other solids using integration.

Read:
pgs. 422 - 425 (through example 2)
pgs. 426-427
pgs. 428 - 429 (examples 7, 8)

Definitions/ Formulas:
Definition of Volume

Flashcards:
Definition of Volume

Examples

Ex1. Volume of a solid of revolution (pg. 424 ex2)

student example

Ex2. Volume of hollow objects (pg. 426 ex4)


Ex3. Volume of shifted hollow objects (pg. 426 ex5)


student example

Ex4. Volumes of other solids

student example


Problem Set: #1-4, 11, 14, 27, 30, 49, 58

6.1b: The Area Between Two Curves

Read:
Pgs. 418-419

Definitions/Formulas:
Definition 3 of the Area Between Two Curves

Flashcards:
Definition 3 of the Area Between Two Curves

Examples:

Ex1. Area between intersecting functions.

Student Example


Ex2. Integrating with respect to the y-axis.

Student Example


6.1b Problem Set:
# 2, 19, 20, 23, 29

6.1a: Area Between two Curves

Objectives:

• Determine the area betwen the graphs of two functions.

Read:

• pgs. 415 - 417

Definitions & Formulas:

• Sigma definition of the area between two curves.

• Integral definition for the area between two curves.

Flashcards:

• Integral definition for the area between two curves.

Examples:


Explanation of the Area Between Two Curves


Ex1. Basic area between two curves.


Student example



Ex2. Using your calculator to establish limits.



Student example.


6.1a problem set:
#5, 6, 12, 33, 36, 41