Solid of revolution

 Topics in calculus Differentiation Integration

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 πr2w units, is enclosed.

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Formulas for solids of revolution

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

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

If one of the bounding curves is actually the x-axis, then we can let [itex]g(x) = 0[itex] in the formula above, and we have:

[itex]V = 2\pi \int_a^b x f(x) dx[itex]

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

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

As above, we can use

[itex]V = \pi \int_a^b [f(x)]^2 dx[itex]

if one of the bounding curves is actually the x-axis.

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.

Disc method

This is used when the slice that was drawn is perpendicular to the axis of revolution. In this case, R(x) is a function that represents the furthest distance between the area and the axis of revolution (typically the end of the slice that is furthest from the axis of revolution), and r(x) is a function that represents the smallest distance between the area and the axis of revolution (typically the end of the slice closest to the axis of revolution):

[itex]V = \pi \int_a^b ([R(x)]^2-[r(x)]^2) dx[itex]

To visualize how this works, consider a function like [itex]Y = e^x[itex], on the interval [0,4] being revolved about the x-axis. If you imagine looking at the graph from the side (so that you are right behind the y-axis) and see the representative slice being revolved about the x-axis, it would form a circle, the area of which is [itex]\pi R^2[itex]. Summing up every one of the areas of the circles (i.e. the definite integral) gives you the the total volume. This is a special case of the Disc method, where r(x)=0.

Shell method

This is used when the slice that was drawn is parallel to the axis of revolution. For this formula, p(x) is a function that represents the distance from a slice to the axis of revolution, and h(x) is a function that represents the height of a given slice:

[itex]V = 2\pi \int_a^b p(x)*h(x) dx[itex]

To visualize how this works, consider the same function and bounds as before, but this time being revolved about the y-axis. If you look at it from above and revolve the slice around the y-axis, it forms a cylinder with no top or bottom. The lateral surface area of any cylinder is given by [itex]2\pi ph[itex], where p is the radius (just keeping it in terms of the formula), and h is the height. Summing up all of the surface areas along the interval (i.e. the definite integral) gives you the total volume.

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