2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

2017 Volume disc washer cross section (1).notebook March 27, 2017

Attachments

2017 VolumesDisk and Washer Methods.docx

2017 VolumesDisk and Washer Methods.pdf

AP CALCULUS - AB

NOTES and EXAMPLES

DeMalade

Topics: Volume: The Disk & Washer Methods

Area is only one of the many applications of the definite integral. We can also use it to find the volume of a three-dimensional solid. The solids which you will focus on in this unit are those with cross sections that are all circular.

(w) (w)

A 2-dimensional figure ………..which, when rotated about the x-axis becomes a…….3-dimensional figure – a cyclinder.

Recall: The formula for the volume of a cylinder is . A disk is basically a very short cylinder where the height, h is actually the w value. The radius retains its actual meaning. The pops out in front and you get the following formula below:

The Disk Method

To find the volume of a solid of revolution with the disk method, use one of the following:

Horizontal Axis of RevolutionVertical Axis of Revolution

(d)

(c)

a b

Example 1: Using the Disk Method.

Find the volume of the solid formed by revolving the region bounded by the graphs of

and the x-axis about the x-axis .

Sketch the graph and draw a representative rectangle.

Example 2: Revolving About a Line That Is Not A Coordinate Axis.

Find the volume of the solid formed by revolving the region bounded by the graphs of

and about the line

Sketch the graph and draw a representative rectangle.

The Washer Method

To find the volume of a solid of revolution with the washer method, use one of the following:

Horizontal Axis of RevolutionVertical Axis of Revolution

(d)

(c)

a b

Example 3: Using the Washer Method.

Find the volume of the solid formed by revolving the region bounded by the graphs of

and about the x-axis.

Sketch the graph and draw a representative rectangle.

Example 4: Integrating with Respect to y, Two Integral Case.

Find the volume of the solid formed by revolving the region bounded by the graphs of about the y-axis.

Sketch the graph and draw a representative rectangle.

Think of the above example in some other ways.

What would your integration set-up look like if you were to revolve the region about

(a) the line x =–1?(b) the line y = -2?(c) the line x = 2?

Advice

1. Draw a representative rectangle that begins from the axis of revolution to the farthest boundary.

(This distance will represent R)

2. Draw a representative rectangle that begins from the axis of revolution to the nearest boundary.

(This distance will represent r)

? Is the “slice” solid? i.e. touches the axis of revolution at all times? Use disc method

? Does the “slice” have a hole in it? i.e. does the original graph touch the axis of revolution at all times?

Use the washer method

AP CALCULUS

NOTES and EXAMPLES

DeMalade

Topics: Volume: Using Known Cross Sections

Solids With Known Cross Sections

(Volumes of Solids with Known Cross Sections1. For cross sections of area A(x) taken perpendicular to the x-axis. 2. For cross sections of area A(y) taken perpendicular to the y-axis. )

Example 5: Triangular Cross Sections.

Find the volume of the solid in which the base is the region bounded by the graphs of

a. The cross sections perpendicular to the x-axis are squares.

b. The cross sections perpendicular to the x-axis are semi-circles.

c. The cross sections perpendicular to the x-axis are equilateral triangles.

2

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Volume ()

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SMART Notebook

c

d

w

w

 AP  CALCULUS  -‐  AB   NOTES  and  EXAMPLES   DeMalade  

  Topics:    Volume:    The  Disk  &  Washer  Methods                                                     Area is only one of the many applications of the definite integral. We can also use it to find the volume of a three-dimensional solid. The solids which you will focus on in this unit are those with cross sections that are all circular. A 2-dimensional figure ………..which, when rotated about the x-axis becomes a…….3-dimensional figure – a cyclinder. Recall: The formula for the volume of a cylinder is . A disk is basically a very short cylinder where the height, h is actually the w value. The radius retains its actual meaning. The pops out in front and you get the following formula below: The Disk Method To find the volume of a solid of revolution with the disk method, use one of the following: Horizontal Axis of Revolution Vertical Axis of Revolution

a b Example 1: Using the Disk Method. Find the volume of the solid formed by revolving the region bounded by the graphs of and the x-axis about the x-axis . Sketch the graph and draw a representative rectangle.

Example 2: Revolving About a Line That Is Not A Coordinate Axis. Find the volume of the solid formed by revolving the region bounded by the graphs of and about the line Sketch the graph and draw a representative rectangle.   The Washer Method To find the volume of a solid of revolution with the washer method, use one of the following: Horizontal Axis of Revolution Vertical Axis of Revolution

a b Example 3: Using the Washer Method. Find the volume of the solid formed by revolving the region bounded by the graphs of and about the x-axis. Sketch the graph and draw a representative rectangle.

c

d

Example 4: Integrating with Respect to y , Two Integral Case. Find the volume of the solid formed by revolving the region bounded by the graphs of

about the y-axis. Sketch the graph and draw a representative rectangle. Think of the above example in some other ways. What would your integration set-up look like if you were to revolve the region about (a) the line x =–1? (b) the line y = -2? (c) the line x = 2? Advice 1. Draw a representative rectangle that begins from the axis of revolution to the farthest boundary. (This distance will represent R) 2. Draw a representative rectangle that begins from the axis of revolution to the nearest boundary. (This distance will represent r) ? Is the “slice” solid? i.e. touches the axis of revolution at all times? Use disc method ? Does the “slice” have a hole in it? i.e. does the original graph touch the axis of revolution at all times? Use the washer method

Volumes  of  Solids  with  Known  Cross  Sections  1.    For  cross  sections  of  area  A(x)  taken  perpendicular  to  the  x-‐axis.  

                                                                       

   2.    For  cross  sections  of  area  A(y)  taken  perpendicular  to  the  y-‐axis.  

                                                                     

AP  CALCULUS   NOTES  and  EXAMPLES   DeMalade      

Topics:    Volume:    Using  KNOWN  CROSS  SECTIONS                                                          

Solids With Known Cross Sections

Example 5: Triangular Cross Sections. Find the volume of the solid in which the base is the region bounded by the graphs of

a. The cross sections perpendicular to the x-axis are squares. b. The cross sections perpendicular to the x-axis are semi-circles. c. The cross sections perpendicular to the x-axis are equilateral triangles.

SMART Notebook

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