Volumes of Revolution

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Volumes of Revolution

• Day 3
• Cylinders (Shells)

2

• Sometimes it isnt easy to get a washer or disk
  when you rotate a curve and slice perpendicular
  to axis of rotation it doesnt work
  mathematically. Think about this

y sinx
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What would the washer look like?

• This is the washer from a side view they stack
  dy so the integration would have to be in terms
  of y.
• The problem is R sin-1y AND r sin-1y

4

• Washers wont work when the two sides of the
  curve are the same function. So what to do?
• There is a shape made parallel to the axis of
  rotation. We call them Shells or Cynlinders.

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This is the general picture
What is the volume of a cylinder?

• Lets see what it looks like in animation

Lets see what it looks like in animation
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What is the areas of each shell?

• In general the area of each shell would be
• You will determine both r and h.
• One will be in terms of x the other in terms of
  y. THEN you will substitute one variable for the
  other and integrate.

7
Remember

• These shells will nest parallel to the axis of
  rotation. So, the stacking will be along that
  axis therefore you will integrate in terms of
  that variable.
• There will be examples when multiple methods
  work. Use what you prefer or recognize first.
  Tonight, however, try only shells

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Examples

• Find the volumes of the figures created
• 1. , , rotated
  around the y axis.
• , y 0, x 4, rotated around the y
  axis.
• 3. , y 0, x 4, rotated around the
  x axis. (AS cylinders)

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Reminder

1. Cylinders will be parallel to axis of rotation
   (new concept)
2. Cylinder nesting will tell you whether you
   integrate in terms of y (dy) or x (dx).