Examples of revolution in the following topics:
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- 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.
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- If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ 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 $A_y$ is given by the integral:
- Use integration to find the area of a surface of revolution
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- 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).