Disk and shell methods compute volumes of rotationally symmetric parts.
Disk method: V=π∫r(x)²dx. Washer method subtracts an inner radius. Shell method: V=2π∫x f(x)dx. They compute volumes of shafts, tanks and vessels from a rotated profile.
Rotating y=x from 0 to 1 gives a cone of volume π/3.
Many machine parts are solids of revolution, so these methods compute their volume, mass and centre of mass quickly.
Disk and shell methods compute volumes of rotationally symmetric parts.
Many machine parts are solids of revolution, so these methods compute their volume, mass and centre of mass quickly.
The disk method slices perpendicular to the axis of rotation. The shell method uses cylindrical shells parallel to the axis and is often simpler when slicing is awkward.
Revolving a semicircle about its diameter gives a sphere, and integrating the disk areas over the radius recovers the familiar four-thirds pi r cubed.
Typical applications include computing the volume of a turned shaft from its profile, finding the capacity of a tank with a curved wall and estimating the mass of a rotating flywheel from its radius profile.
Study "Definite integrals and area" first (topic 35 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Rotate about x or y axis. Choose the method with simpler radius. Pappus theorem links volume to centroid.
Rotating y=x from 0 to 1 gives a cone of volume π/3.
The next topic is "Centroids and moments", which builds directly on this one.
It is classed as Beginner in Engineering Mathematics (topic 36 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
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