Solid of revolution

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In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane.

Assuming that the figure lies entirely on one side of the axis, the solid's volume is equal to the length of the circle described by the figure's barycenter, times the figure's area.

A representative disk is three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length "w") around some axis (located "r" units away); such that, a cylindrical volume, of π∫r²w units, is enclosed.

Rotating a curve

1 Methods of finding volume: disc and shell methods
- 1.1 Disc method
- 1.2 Shell method
2 See also
3 External links

[edit] Methods of finding volume: disc and shell methods

With these methods, it is easiest to draw the graph(s) in question, identify the area that is actually being revolved about the axis of revolution, and then draw a straight line, vertical for functions defined in terms of x and horizontal for functions defined in terms of y, which is referred to as a slice. Note that although all formulas are listed in terms of x, the formulas are exactly the same for functions defined in terms of y (with rotations about the x- and y-axes appropriately reversed).

[edit] Disc method

This is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when you are integrating along the axis of revolution.

The volume of the solid formed by rotating the area between the curves of $f (x)$ and $g (x)$ and the lines $x = a$ and $x = b$ about the x-axis is given by

$V = \pi \int_a^b [f(x)^2 - g(x)^2]\,dx$

If g(x) = 0 (e.g. revolving an area between curve and x-axis), this reduces to:

$V = \pi \int_a^b f(x)^2\,dx$

To visualize how this works, consider a thin vertical rectangle at x between $y = f (x)$ on top and $y = g (x)$ on the bottom, and revolve it about the x-axis; it forms a ring (or disc in the case that $g (x) = 0$ ), with outer radius f(x) and inner radius g(x). The area of a ring is $π(R 2 - r 2)$ , where R is the outer radius (in this case f(x)), and r is the inner radius (in this case g(x)). Summing up all of the areas along the interval gives you the total volume.

[edit] Shell method

This is used when the slice that was drawn is parallel to the axis of revolution; i.e. when you are integrating perpendicular to the axis of revolution.

The volume of the solid formed by rotating the area between the curves of $f (x)$ and $g (x)$ and the lines $x = a$ and $x = b$ about the y-axis is given by

$V = 2\pi \int_a^b x[f(x) - g(x)]\,dx$

If g(x) = 0 (e.g. revolving an area between curve and x-axis), this reduces to:

$V = 2\pi \int_a^b x f(x)\,dx$

To visualize how this works, consider a thin vertical rectangle at x with height $[f (x) - g (x)]$ , and revolve it about the y-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is $2π r h$ , where r is the radius (in this case x), and h is the height (in this case $[f (x) - g (x)]$ ). Summing up all of the surface areas along the interval gives you the total volume.