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
Arc Length, Surface Area and Polar Coordinates
Polar Coordinates & Parametric Equations
My Belief Statement
My Belief Statement: _It is better to be defeated on principle than to win on lies._ Arthur Calwell
Many students have a serious phobia when it comes to Math class, let alone writing an exam in that subject. I believe this phobia can be dispelled by "digging" into the "WHY " and the "HOW". In fact, many students WANT this. I attempt, wherever possible, to connect a geometric representation to its algebraic counterpart; an analysis of how the two versions interact is important. Contrary to what SOME individuals may think, I'm not at all afraid of technology. I AM, however, cautious to not "jump into the deep end"; we need to grow with this and "dovetail" the old with the new so that they COMPLIMENT each other.
I have several links in the sidebar at the right, as well as some of my own discriptions on various topics. These will grow in number and evolve over time, as will my hand-written notes; these can be viewed by following appropriate links which you will find throughout this blog. These notes will also be posted on my facebook site if that happens to be your preferred mode of viewing; I've included a link to that in the sidebar as well.