Project #1 by Maya Elias

For my project, I decided to use the washer method. This method is similar to the disk method in that it is used to calculate the volume of a graph around the x- or y-axis, but this method is useful when the graph has a hole in it. Just like a washer (pictured to the right) that you'd find in a tool kit, our 3D image can be visualized as a bunch of different sized washers stacked on one another.

I chose to use the washer method because it worked best with my particular functions. I chose the functions

\[f(x) = (1/10) * x^2\] and \[f(x) = 0\]

because I wanted my 3D printed image to look like a bowl. I figured a bowl could be useful and cute and still properly demonstrate the washer method.

The 3D model I created(pictured on the left) is made up of mulitple hollow cylinders and one solid cylider in the center. Because of its composition, in order to calculate the volume of the solid, I calculated the volume of each individual cylinder minus the section that was removed. The following work shows how my calculation was completed:





\[V = pi*r^2*h\] \[ V = pi ( (1^2(1/10)) + (2^2(2/5)-1^2(2/5)) + (3^2(9/10)-2^2(9/10)) + (4^2(8/5)-3^2(8/5)) \] \[+ (5^2(5/2)-4^2(5/2)) + (6^2(18/5)-5^2(18/5)) + (7^2(49/10)-6^2(49/10)) + (8^2(32/5)-7^2(32/5)) \] \[+ (9^2(81/10)-8^2(81/10)) + (10^2(10)-9^2(10)) = 1779.71 units^2 \]

The washer method tells us that in order to calculate the volume of the solid you must integrate the area of the cylinder created minus the area of the cylinder removed. The actual volume of the solid you create when rotating my two equations about the y-axis from [0,10] is shown in the calculation below:

Clearly, the estimated volume is slightly above the actual volume, but an overestimate is to be expected because the cylinders I chose to use for my estimate all went slightly above the actual function. The more cylinders I would've used and the smaller the interval I chose, would have created a closer estimate.

My 3D printed object will be scaled down so that the radius of my final project is approximately 1 inch and the height is approximately 1 inch at its tallest point. This object is project to take about 1.5 hours to print.