• Discussion [Using Flash] [Using Java]

  • Finding the volume of a right circular cone.
    Discussion [Using Flash] [Using Java]

  • An animation showing the rotation of a graph about the x-axis.

  • More details on the solid obtained in the animation above.

  • Finding the volume of a rotated graph.
    Discussion [Using Flash] [Using Java]

  • Examples:

    Find the volume of the solid obtained by rotating the following areas about the x-axis:

    1. Area bounded by the graph of f(x) = x - x2 and the x-axis.
      Solution [Using Flash] [Using Java]
      [Graph]

    2. Area bounded by the graphs of f(x) = x3 - x + 1, x = -1, x = 1 and the x-axis.
      Solution [Using Flash] [Using Java]
      [Graph]

  • Finding the volume obtained by rotating the area between the graphs of two functions about the x-axis.
    Discussion [Using Flash] [Using Java]

  • Examples:

    Find the volume of the solid obtained by rotating the following area about the x-axis:

    1. Area bounded by the graphs of f(x) = x2 and g(x) = x3.
      Solution [Using Flash] [Using Java]
      [Graph]

    Find the volume of the solid obtained by rotating the area bounded by f(x) = x2 and g(x) = x about the following lines:

    1. y = -1.
      Solution [Using Flash] [Using Java]
      [Graph]

    2. y = 2.
      Solution [Using Flash] [Using Java]
      [Graph]

    3. y-axis.
      Solution [Using Flash] [Using Java]
      [Graph]