Volumes

1
Volumes

• 7.3

2
Methods of Finding Volume

• Known cross sections a solid is made by stacking
  squares or equilateral triangles or circles or
  isosceles triangles.
• Circular Cross sectionsrevolving a shape around
  an axis where the shape is attached to the axis.
• Washer sectionsrevolving a shape around an axis
  where there is a space between the shape and the
  axis.

3
Methods of Finding Volume

• Shell methodcan be used instead of either washer
  or circle methods for two reasons
• cuts down the number of integrals
• Inability to solve for a variable as needed
• For a visual of this method follow the link
• http//mathdemos.gcsu.edu/mathdemos/hyperlist/cata
  log.htmlV

4
Methods of Finding Volume

• Known cross sections a solid is made by stacking
  squares or equilateral triangles or circles or
  isosceles triangles.
• If the cross sections are perpendicular to the
  x-axis or a horizontal axis, then dx will be the
  thickness. Find the formula for the area in
  terms of x and integrate.
• If the cross sections are perpendicular to the
  y-axis or vertical axis, then dy will be the
  thickness. Find the formula for the area in
  terms of y and integrate.

5
Methods of Finding Volume

• Circular Cross sections
• When revolving around the x axis or horizontal
  axis the circles will be vertical. So the
  thickness of the circles will be dx. Volume is
  circles piled up with thickness dx so xs. Find
  the area of the circles in terms of x and
  integrate. The limits go from left-most x to
  right-most x of your shape. The shape must
  directly connect to the x-axis.

6
Methods of Finding Volume

• Circular Cross sections
• When revolving around the y axis or vertical axis
  the circles will be horizontal. So the thickness
  of the circles will be dy. Volume is circles
  piled up with thickness dy so ys. Find the area
  of the circles in terms of y and integrate. The
  limits go from lowest y to highest y of your
  shape. The shape must directly connect to the
  y-axis.

7
Methods of Finding Volume

• Washer Cross sections
• When revolving around the x axis or horizontal
  axis the washers will be vertical. So the
  thickness of the washers will be dx. Volume is
  washers piled up with thickness dx so xs. Find
  the volume of the outer curve as if attached to
  the axis, then repeat for the inner curve and
  subtract. The limits go from left-most x to
  right-most x of your shape.

8
Methods of Finding Volume

• Washer Cross sections
• When revolving around the y axis or vertical axis
  the washers will be horizontal. So the thickness
  of the washers will be dy. Volume is washers
  piled up with thickness dy so ys. Find the
  volume of the outer curve as if attached to the
  axis, then repeat for the inner curve and
  subtract. The limits go from lowest y to highest
  y of your shape.

9
Methods of Finding Volume

• Shell method
• Horizontal Axis of revolution makes the cylinders
  horizontal so the thickness is dy. Therefore,
  find 2prh in terms of y and integrate from the
  lowest y to the highest.
• Vertical axis of revolution makes the cylinders
  vertical, so the thickness is dx. Therefore,
  find 2prh in terms of x and integrate from the
  lowest x to the highest.

10
Examples

• Handout of examples will be given out in class.