Examples of revolution in the following topics:
-
- Shell integration (also called the shell method) is a means of calculating the volume of a solid of revolution when integrating perpendicular to the axis of revolution .
- (When integrating parallel to the axis of revolution, you should use the disk method. ) While less intuitive than disk integration, it usually produces simpler integrals.
- By adding the volumes of all these infinitely thin cylinders, we can calculate the volume of the solid formed by the revolution.
- Each segment located at x, between f(x)and the x-axis, gives a cylindrical shell after revolution around the vertical axis.
- Use shell integration to create a cylindrical shell and calculate the volume of a "solid of revolution" perpendicular to the axis of revolution.
-
- If the curve is described by the function $y = f(x) (a≤x≤b)$, the area Ay is given by the integral Ax=2π∫abf(x)√1+(f′(x))2dx for revolution around the x-axis.
- A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis .
- If the curve is described by the parametric functions x(t), y(t), with t ranging over some interval [a,b] and the axis of revolution the y-axis, then the area Ay is given by the integral:
- Use integration to find the area of a surface of revolution
-
- Disc and shell methods of integration can be used to find the volume of a solid produced by revolution.
- The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution.
- The shell method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.
- Integration is along the axis of revolution (y-axis in this case).
- The integration (along the x-axis) is perpendicular to the axis of revolution (y-axis).