Integration – Surface Areas & Volumes of Revolution


When the graph of a 2-dimensional function is revolved about a vertical or horizontal axis, the result is referred to as a 3-dimensional solid.  By applying some of the concepts we have already learned, the surface areas and volumes of many such solids of revolution can be determined with relative ease.  The notes contained in the link directly below call on our knowledge of arc length to assist in determining the area of surfaces of revolution; direct reference is made here to the “onion proof” that was introduced in the previous set of notes.  The premise behind this approach is to find the sum of the areas of infinitely many, infinitely narrow “bands” that are “wrapped” around the surface of revolution.  If one of these “bands” was cut and then stretched out, it would form a rectangle whose length would be equivalent to the circumference (arc length) of the surface of revolution at that point; the width of this “band” would ultimately be represented by “dx” or “dy”, depending on the axis of revolution. 

Several examples involving the calculation of surface areas are illustrated in my notes; following those are two methods of determining volumes of solids of revolution, the first being the “Shell Method”.  This method once again reflects back on the “onion proof” that was used previouly to derive the formula for area of a circle.  The idea behind the “onion proof” is that “layers” (bands) are added uniformly around the circumference until the desired radius is attained.  The areas of these infinitely narrow “layers” are then added to determine the area of the circle; this circle essentially forms the base of the cylindrical shells that emerge as a result of revolving a function about an axis.  The “height” of these cylindrical shells is determined by the function itself and is always perpendicular to its circular base; the volume of each cylindrical shell is the product of its “height” and the area of its circular base.  As layers are added to the circular base, the “height” of these cylindrical shells also changes, governed by the function itself; the volumes of these cylindrical shells, having infinitely thin lateral surfaces, are ultimately added to determine the volume of the solid itself.  After working through several examples using the Shell Method,  the volumes of many of the same solids are calculated using the Slab (Disk) Method.  With this method, cylindrical “slabs” of infinitely small “height” are stacked together, their individual volumes added to determine the volume of the solid of revolution.

Surface Areas & Volumes of Revolution – Samuelson

Additional links have been included below that will reinforce (and go beyond) the notions presented above.

Area of a Surface of Revolution

Volumes of Revolution


The Ellipsoid

Arc Length, Surface Area and Polar Coordinates

Polar Coordinates & Parametric Equations

  1. […] Samuelson has explanations and notes about Integration – Surface Areas & Volumes of Revolution posted at samuelson […]

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