Solids of Revolution : Volumes by Cylindrical Shells : Eggs and Shells

Eggs and Shells

An object of revolution is an object that is formed by taking the area under a curve(s) and rotating it about an axis. The volumes of such objects can be difficult to calculate; while others may be easier. Some objects whose volumes would be easy to calculate include a basketball, can of soup, or an icecream cone as these objects have known volume formulas—i.e. a sphere, cylinder, and cone, respectively. Other objects can prove to more difficult in terms of finding their volumes; some of these objects might include a baseball bat, door knob, or doughnut. We will be looking at finding the volume of an egg, which is similar to a sphere but has more of an irregular oval or spheroid shape—in that it is 'fatter' at the bottom—but first a discusion on the method of volumes by cylindrical shells.

The method of volumes by cylindrical shells takes an object of revolution and breaks in down into concentric cylinders of constant thickness, called shells. The object can either be hollow, like in the case of a doughnut, meaning that it has no solid center and that the innermost cylinder is also a shell; or the object can be solid, as in the case of the baseball bat or door knob, such that the innermost cylinder is solid. Furthermore, these shells are of variable height such that the height of the cylinders follows the curve(s) that were revolved around an axis to create the object. These shells have easy to compute volumes which allows us to approximate the volume of the object itself. The volume of one of the shells can be determined via the following formula \[ V = 2 \pi r h \Delta r \] where \(r\) is the average radius of the shell given by \( \displaystyle r = \frac{1}{2} ( r_2 + r_1 ) \), \(h\) is the height if the cylinder, and \( \Delta r \) is given by \( \Delta r = r_2 - r_1 \). Note that this formula for the volume of a shell take into account the fact that it is a shell, rather than a solid cylinder, and removes the volume of the hollow interior. In order to approximate the volume of an object we can sum the volumes of the shells. The more shells you have the more accurate your approximation will be as there will be less loss (or gain) resulting from where the shell meets the curve(s)—this loss (or gain) would result from the shell meeting the curve(s) at either its outside edge, inside edge, or at its midpoint.

Knowing that our approximation increases in accuracy as the number of shells increases it follows that if we had an infinte number of shells we would obtain the actual voulume of our object. Thus it follows that the volume of on object of revolution can be calculated using the following integral: \[ V = \int_{a}^b 2 \pi x f(x) dx \] where \( f(x) \) is the curve(s) that defines the object.

The object we will be both approximating and calculating the volume of will be an egg. The formula \[ y = \pm \frac{B}{2} \sqrt{\frac{ L^2 - 4 x^2 }{ L^2 + 8 w x + 4 w^2 }} \] where, \( L \) is the maximum length of the egg, \(B\) is the maximum breadth, and \( w = \frac{ L - B }{2n} \) is a parameter that shows the distance between two verticle lines corresponding to the maximum breadth and half the length of the egg,plots a curve that is very similar to the curve of an egg. After measuring an egg from my refrigerator if found that \( L = 2" \approx 50 \text{mm} \) and \( B = 1.5" \approx 38 \text{mm} \); I also chose \( n = 2 \) to give my curve the desired egg shape. Observe how we went from an actual egg to a graph of its curvature to a digital 3D approximation.

Now we will compute the volume of our shells and their sum in order to find an approximation of the volume of the original egg. A diagram of a cross section of our 3D approximation showing our actual shells as well as a table detailing their radii, heights, volumes, and the approximations total volume can be found below.

The volumes of the shells were determined using the formula for shell volume defined above. This resulted in an approximate egg volume of 43,803 cubic millimeters or 43.8 milliliters. We can now compute the value of the integral given the function for our curve, the volume formula, and our parameters. \[ V = \int_{-25}^{25} \frac{38}{2} \sqrt {\frac{ 50^2 - 4 x^2 }{ 50^2 + 8 * 3 x + 4 3^2 }} \] This yeilds a volume of 50,561 cubic millimeters or 50.5 milliliters, which means that our approximation was only off by 6.7 milliliters. Therefore, the method of cylindrical shells in a fairly accurate way to approximate the volume of an object of revolution.