Volumes in Calculus

Doug Kuhlmann, SAVAŞ ORHAN, Mar 19, 2018

Volumes: Examples of solids of revolution using disks and shells and solids with known cross sectional areas. Probably works better when downloaded than running in browser.

Table of Contents

  1. Disks and washers
    1. Volume of region rotated around x-axis
    2. Region between two quadratics rotated about y=-3
  2. Cylindrical shells
    1. Region bounded by two quadratics rotated about x=2
    2. Volume generated by rotating region about x=2
  3. Known cross sectional areas
    1. Volume: Circular base cross sections isos rt triangles
    2. Volume: Circ base cross section isos rt triangle hyp in base
    3. Volume: Base between 2 quad, cr. sect. right triangles.
    4. Volume: Hexagons with side in base
    5. Volume: Hexagon cross sections with diagonal in region
    6. Volume: Hexagon2 cross section diagonal in region
  4. Intersecting cylinders
    1. Volume of intersecting cylinders
    2. Volumes of intersecting cylinders of radius 2

1.

Disks and washers

  • 1. Volume of region rotated around x-axis
  • 2. Region between two quadratics rotated about y=-3

1.1.

Volume of region rotated around x-axis

– Doug Kuhlmann

1.2.

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2.

Cylindrical shells

  • 1. Region bounded by two quadratics rotated about x=2
  • 2. Volume generated by rotating region about x=2

2.1.

Region bounded by two quadratics rotated about x=2

– Doug Kuhlmann

2.2.

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3.

Known cross sectional areas

  • 1. Volume: Circular base cross sections isos rt triangles
  • 2. Volume: Circ base cross section isos rt triangle hyp in base
  • 3. Volume: Base between 2 quad, cr. sect. right triangles.
  • 4. Volume: Hexagons with side in base
  • 5. Volume: Hexagon cross sections with diagonal in region
  • 6. Volume: Hexagon2 cross section diagonal in region

3.1.

Volume: Circular base cross sections isos rt triangles

Slide 'a' to move the triangle. Turn on Traces to see the solid. Can also show the surface of the volume.
– Doug Kuhlmann

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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4.

Intersecting cylinders

  • 1. Volume of intersecting cylinders
  • 2. Volumes of intersecting cylinders of radius 2

4.1.

Volume of intersecting cylinders

Shows intersection to two cylinders that have the same radius and whose axes are at right angles. User can change the radius from r=1 to r=5.
Volume of intersecting cylinders
– Doug Kuhlmann

4.2.

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