Introduction
Over the semester we've looked at many topics and created 3D models. For this we are going to revisit an old topic, double integrals over a region. In this we found the volume of a surface in the xyz-plane bounded by two curves. From the many topics I chose to revisit this topic. I have a couple reason to why I chose to redo this. First, the model did not print correctly. The print added spaces between the rectangular prisms. Another reason was that I think the surface and curves did not represent the topic entirely. The surface I chose just increased between the curves.
Improvements
When making the model on Onshape there were no spaces between the rectangles, which can be seen on the right. However, when printing this spaces were being added.
The second issue was with the surface I chose which was \(f(x,y)=xy+x\). This function only increased over the two curves I chose \begin{align*} f(x) &= \sqrt{x} \\ g(x)&= x^3 \end{align*} Having only increasing values can mislead learners on how this topic works. To improve from this, I thought of using a surface that can show increases and decreases in the bounded area that would be selected. In the previous example I used rectangular prisms, which could have been the reason to cause the spaces when being printed. To avoid this problem, I will be using squares instead.
New Example
The surface I chose to do this time was \(f(x,y)=\frac{3}{x^2+1}+\sqrt x\). Bounded between \( \left[-1.2,0.8\right] X \left[1,3\right]\). This time I am bounding the surface by a rectangle instead of curves. This allows us to focus on the idea I am trying to show. Using these bounds we will be able to see increases and decreases for the surface. In the figure below we can see how the model would look with the approximated volumes. The approximated volume was found to be 14.674 units cubic, and the actual volume was 14.8996 cubic units.
In the picture above we can see the increase and decrease of the volume for each square. While in the first example there were only increases. By using squares this time there should be no issues in the printing process. I can see the improvements that I wanted in this example and still get the approximated volume close to the actual volume. When printed the base of the model should be 2in. X 2in.
Over the semester we've looked at many topics and created 3D models. For this we are going to revisit an old topic, double integrals over a region. In this we found the volume of a surface in the xyz-plane bounded by two curves. From the many topics I chose to revisit this topic. I have a couple reason to why I chose to redo this. First, the model did not print correctly. The print added spaces between the rectangular prisms. Another reason was that I think the surface and curves did not represent the topic entirely. The surface I chose just increased between the curves.
Improvements
When making the model on Onshape there were no spaces between the rectangles, which can be seen on the right. However, when printing this spaces were being added.
The second issue was with the surface I chose which was \(f(x,y)=xy+x\). This function only increased over the two curves I chose \begin{align*} f(x) &= \sqrt{x} \\ g(x)&= x^3 \end{align*} Having only increasing values can mislead learners on how this topic works. To improve from this, I thought of using a surface that can show increases and decreases in the bounded area that would be selected. In the previous example I used rectangular prisms, which could have been the reason to cause the spaces when being printed. To avoid this problem, I will be using squares instead.
New Example
The surface I chose to do this time was \(f(x,y)=\frac{3}{x^2+1}+\sqrt x\). Bounded between \( \left[-1.2,0.8\right] X \left[1,3\right]\). This time I am bounding the surface by a rectangle instead of curves. This allows us to focus on the idea I am trying to show. Using these bounds we will be able to see increases and decreases for the surface. In the figure below we can see how the model would look with the approximated volumes. The approximated volume was found to be 14.674 units cubic, and the actual volume was 14.8996 cubic units.
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