Putting the Cube in the Circle Hole

Circles aren't the only shape that can take a stab at guessing the volume of a solid. Any shape with the an area that can be calculated can be used to approximate the area of a solid. The method of approximation by cross section takes advantage of a certain shape to generate a model of 3-D structure. This concept is a more general form of the previously explored disks and washers. This time around, one function will be used as bound for a region that will be rotated around an axis. That function is: \[ f(x) = tanx \] bounded at the line y = 1.

Approximation Using Squares

For this example, squares will be used as the cross sections to create the approximation. The reason that this function was chosen was because it seemed like an interesting shape to rotate around the x-axis. It also seemed ammusing to utilize squares to approximate a trig function. It almost feels like trying to push a cube through a circle shaped hole, but maybe its approximation will be excellent.

Creating the Squares and Rectangular Prisms

The process of creating the squares is quite similar to the washer method for solids of revolution. Rectangular prisms of various widths and lengths are stacked on top of each other to create a rough outline of what the solid is supposed to look like. The length of the prism is explicitly defined by the function of interest; the width is equal to the length since the approximation is performed using squares. The prisms all have one dimension that is identical: the depth. This model will have 10, right endpoint based layers. The first prism has length and width defined by the function at x = .1, and the last is defined by the value at x = 1. This value determines half of the side length of the cross sectional square, rather than the full side. The following table shows the values for all the endpoints:

This table defines the dimensions for all 10 of the rectangular prisms that will make up the model, and here is a picture of the 3D approximation.

The highlighted prism is the 10th and largest layer of the model.

Using the Model to Approximate Volume

Now that the model is constructed, there is now a way to compute a volume that approximates the solid created by the rotation of the image. To calculate the model's volume, the volume if each individual prism has to be calculated and summed. The volume for a prism is: \[ V = L*W*D \] As mentioned earlier, these values are defined by the function and the number of partitions. This table shows the dimensions of each layer and the volume of that layer:

This comes out to a total approximated volume of 2.75 cubic inches.

Finding the Actual Volume of the Solid

In this case finding the accuracy of the approximation is actually quite easy. The function simply has to be integrated with respect to the x-axis. Since the one function defines both the length and width of dx, f(x) needs to be squared in the interval and it is all set up. One thing to note is to get the volume of the whole solid, the function actually needs to be 2f(x) since f(x) only represents half of the length of the square. The bounds of the integral are [0,1] and looks like: \[V = \int_{0}^{1} (2tanx)^2 \,dx = 4(tanx - x)\Big|_0^1 \ = 2.230\] This means the volume of the solid created by the region is 2.230 cubic inches. The approximation was off by about 23.33%, so the approximation was neither amazing or terrible. This was somewhat expected considering the shape of the cross section and the shape of the function.