Introduction
Volume can be found of anything solid. Knowing how to take the volume of a simple shape can help measure the volume of any shape or solid. How can this be done? Taking a solid and dividing it up into multiple parts, which are the cross-sections, and measuring the volume for each of the cross-sections separately. The cross-sections would be a shape we do know how to take the volume of easily like a square, rectangle, or oval. There is no limit on how many cross-sections can be used. Using more cross-sections can give an accurate and closer volume to what the solid actually is. To take a closer look at this lets look at the equations \[y = 1-x^2\] and \[y = -1+x^2\] When you graph these two functions it seems like a football from \(-1≤x≤1\) as below.
To approximate this solid we will be using 10 rectangle cross-sections perpendicular to the x-axis. The graph above is in 2-dimensional to make this into 3-dimensional imagine cutting this football shaped solid into 10 parts. With these parts imagine rectangles raising out of them where the width is the distance from one point to another for the same x-value and the height is half of the width.
The picture on the left can help visualize how the rectangular cross-sections would look like on the 2-d plane. We can see here as we get closer to the outer edges the rectangles get smaller as we could have predicted from the image from the graph.
With the image on the right, we can see the same model as above, but in a different angle. With this angle we can see the that the rectangles are centered, which makes the shape look like a football.
Compute Volume
We will be computing the volume using the cross-sections and the actual volume. This way we would also be able to see how close the approximation is. To measure the real volume we would use the following formula: \[ V = \int_a^b A(x)dx\] This is the general formula to find the volume for all functions. Since we are using rectangles as the cross-sections A(x) would be the height multiplied by the width and the interval would be from \(-1≤x≤1\). \[V = \int_{-1}^{1} (1-x^2)*((1-x^2)-(-1+x^2)) dx\] This would give the volume of 2.13333 cubic units. This is the actual volume without using cross-sections.
To measure the volume with the rectangle cross-sections we will use the midpoints. By doing this we would be able to divide the interval into 10 pieces and making 10 visible rectangles. This would also make the length of each rectangle the same (0.2).
The table above calculates the volume for each rectangle. The X column are the midpoints. The next two columns shows the y-value for the x point. The width is calculated by adding the absolute value of these y-values. The height was mentioned to be half the width. Then in the last column is the volume. The volume formula used to caluate each rectange is the following \[ V = l*w*h\] For the final volume all the volumes for each rectangle were added together. The final volume was 2.13352 cubic units when we approimated using cross-sections. The real volume was found to be 2.13333 cubic units. We can see here the volume found was a good approximation to the actual volume.
Why these Functions
We could have had the rectangles lay flat against one side which would have gave the same volume but a different shape. By centering the rectangles on top of each other gave a final product that looks like a football. Being able to visualize a certain object and then applying math to it was helpful to me. I chose these functions because the computations were easier which helped to focus on the process rather than the calculus. Taking the midpoints instead of the right or left points also helped with the computations. However, using the midpoints did give us a closer answer to the actual volume rather than using other points. The final 3-d printed object is below. The dimesnions are 2X2 inches.
Volume can be found of anything solid. Knowing how to take the volume of a simple shape can help measure the volume of any shape or solid. How can this be done? Taking a solid and dividing it up into multiple parts, which are the cross-sections, and measuring the volume for each of the cross-sections separately. The cross-sections would be a shape we do know how to take the volume of easily like a square, rectangle, or oval. There is no limit on how many cross-sections can be used. Using more cross-sections can give an accurate and closer volume to what the solid actually is. To take a closer look at this lets look at the equations \[y = 1-x^2\] and \[y = -1+x^2\] When you graph these two functions it seems like a football from \(-1≤x≤1\) as below.
To approximate this solid we will be using 10 rectangle cross-sections perpendicular to the x-axis. The graph above is in 2-dimensional to make this into 3-dimensional imagine cutting this football shaped solid into 10 parts. With these parts imagine rectangles raising out of them where the width is the distance from one point to another for the same x-value and the height is half of the width.
The picture on the left can help visualize how the rectangular cross-sections would look like on the 2-d plane. We can see here as we get closer to the outer edges the rectangles get smaller as we could have predicted from the image from the graph.
With the image on the right, we can see the same model as above, but in a different angle. With this angle we can see the that the rectangles are centered, which makes the shape look like a football.
Compute Volume
We will be computing the volume using the cross-sections and the actual volume. This way we would also be able to see how close the approximation is. To measure the real volume we would use the following formula: \[ V = \int_a^b A(x)dx\] This is the general formula to find the volume for all functions. Since we are using rectangles as the cross-sections A(x) would be the height multiplied by the width and the interval would be from \(-1≤x≤1\). \[V = \int_{-1}^{1} (1-x^2)*((1-x^2)-(-1+x^2)) dx\] This would give the volume of 2.13333 cubic units. This is the actual volume without using cross-sections.
To measure the volume with the rectangle cross-sections we will use the midpoints. By doing this we would be able to divide the interval into 10 pieces and making 10 visible rectangles. This would also make the length of each rectangle the same (0.2).
The table above calculates the volume for each rectangle. The X column are the midpoints. The next two columns shows the y-value for the x point. The width is calculated by adding the absolute value of these y-values. The height was mentioned to be half the width. Then in the last column is the volume. The volume formula used to caluate each rectange is the following \[ V = l*w*h\] For the final volume all the volumes for each rectangle were added together. The final volume was 2.13352 cubic units when we approimated using cross-sections. The real volume was found to be 2.13333 cubic units. We can see here the volume found was a good approximation to the actual volume.
Why these Functions
We could have had the rectangles lay flat against one side which would have gave the same volume but a different shape. By centering the rectangles on top of each other gave a final product that looks like a football. Being able to visualize a certain object and then applying math to it was helpful to me. I chose these functions because the computations were easier which helped to focus on the process rather than the calculus. Taking the midpoints instead of the right or left points also helped with the computations. However, using the midpoints did give us a closer answer to the actual volume rather than using other points. The final 3-d printed object is below. The dimesnions are 2X2 inches.
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