Calculating the volume of "irregular" solids—as opposed to regular solids such as rectangular prisms, a square pyramid, etc.—can be difficult. We have previously reviewed solids of revolution and a method of obtaining the volume of one such solid, specifically the method of volumes by cylindrical shells. This method is useful for finding the volumes of "round" or "circular" solids—i.e. those that have been revolved about an axis; while the formulas for the volumes of spheres and cones are known we can easily approximate their volumes via the washer or shell methods, furthermore, the volumes of solids whose volumes are unknown—like an egg, doorknob, or lampshade—can also be easily approximated using these methods. We are now examining other solids that have not been revolved about an axis or are not "round" such as a roof or an eye glasses case. In order to approximate the volume of solids like these it is necessary for the solid to have a known cross section or, more plainly, a slice parallel to an axis that has a known or easily calculated area. Ideally, the slices from our solid will be regular polygons or ellipses, as that will make approximating volume easier.
Method
An approximation of a solid of known cross section found by "slicing" the solid and summing the volume of each slice is only so accurate. An accurate volume can be found in the same method as for a solid of revolution, integration—this follows as every solid of revolution is technically a solid of known cross section, or solids of revolution are special cases of solids of known cross section. The volume of a solid of known cross section can be found using the following ingteration: \[ V = \int_{a}^b A(x)dx, \] where \( V \) is volume, \(A(x)\) is the area of a cross section of the solid, and \(x\) lies between \(a\) and \(b\). This is the same as if we had infinitely many incredibly thin cross sections summed together, which gives a more accurate measure as each cross section is a much better fit than our approximation. Note that we are slicing along the \(x\)-axis, if we integrated in terms of \(y\) we would be slicing along the \(y\)-axis.
The Best Thing Since Sliced Bread
When thinking of solids that are commonly sliced I immediately thought of bread, but determined that a solid with a simpler cross section would be easier to think about, approximate a volume for using cross sections, integrate to find the actual volume, and model with a 3-D printed object. Thus, I landed on a toaster. So, I found a toaster and measure its length \( (L) \), width \( (W) \), and height \( (H) \); as well as observing its general form; and determined the following measurements, which I then scaled down to make them easier to handle. Those measurements are as follows: \[{ L = 12.8" \rightarrow 1" \\ W = 8" \rightarrow 0.625" \\ H = 9" \rightarrow 0.703" }\] Additionally, we must also consider the shape of the base of the toaster as well as the curve that makes of the top of the toaster. The base of the toaster is the area that lies between the following two curves: \[{ y = (0.5^6 - (x-0.5)^6)^{\frac{1}{6}} - 0.1875 (1)\\ y = -(0.5^6 - (x-0.5)^6)^{\frac{1}{6}} + 0.1875 (2). }\] And the curve that makes up the top of the toaster is the following: \[ y = (0.5^6 - (x-0.5)^6)^{\frac{1}{6}} + 0.203 (3). \] These next images show the toaster I modeled my solid of known cross section, the graph of toaster's base (in red), and the curve that makes up the top of the toaster (in black).
Note that the cross sections of this solid were taken to be rectangles whose length was 0.1" in order to slice the solid 10 times, the width of each slice was determined by adding the outputs of equations (1) and (2) when inputting the \(x\) value that corresponded to that cross sections—i.e. the cross section that occurred at 0.1", 0.2", etc.—and the height of each slice was determined by inputting that same \(x\) value into equation (3). The results of those calculations can be found in the following table, as well as the volume of each cross section, and the sum of the volumes of the cross sections that gives the approximated total volume of the toaster—note, all values in the table have been scaled back to their original values by multiplying by a scalar of 12.8".
Knowing that our cross sections are rectangles as well as the length, width, and height of each cross section allows us to create a 3-D model of our toaster approximation the looks like the following image, with a dimension drawing following that. Note that the two vertical indentions in the model are there to show the slices and separate the cross sections; the volume lost to these indentions is ignored as they have been added later for clarity.
We can now go about finding the actual volume of the solid by performing the following integration, where \(V\) is volume: \[ V = \int_{0}^1 [((0.5^6 - (x-0.5)^6)^{\frac{1}{6}} - 0.1875) + (-(0.5^6 - (x-0.5)^6)^{\frac{1}{6}} + 0.1875)][(0.5^6 - (x-0.5)^6)^{\frac{1}{6}} + 0.203]dx. \] When integrated correctly this integral equals approximately \( \displaystyle 0.4033 \text{ in}^3 \); which must then be scaled back to our original dimensions, this is done by multiplying the above result by \( 12.8^3 \) as each of the three elements of the area calculation must be scaled back to the original, which yields a volume of \( \displaystyle 845.7372 \text{ in}^3 \). This volume makes since as our approximation using known cross sections yielded a volume of \( \displaystyle 805.100 \text{ in}^3 \); which is about \( 95 \% \) of our actual volume—found via integration—and this tells us that the approximation of the volume of a solid of known cross section is really very accurate.
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