How to think about volumes of solids of known cross-section
Solids of known cross-section are not as hard to think about as they seem. Starting with the graph of a two-dimensional function, one can slice it into pieces; for each piece, one can take a known shape in three dimensions and visualize it in place of each slice. Often, it is easier to see than to explain. Consider the example below:
Here, the two-dimensional x- and y-planes lie flat on the table. The function graped has been sliced into pieces of equal height. Each slice has then been represented in three dimensions as a semicircle. The only "bad" thing about this image is that it does not clearly show the thickness of each semicircle which is needed to calculate the volume of each cross-section. For our purposes, this image works because it gets us thinking about what we're going to do next as we learn how to visualize and compute said volumes.
For simplicity's sake, we are going to consider the solid bounded by quadrant I and the function \(y = \sqrt{100-x^2}\) seen below where every cross-section perpendicular to the x-axis is a square.
Why I chose this solid
This solid works well for beginners because the first area formula we are taught growing up is the area of a square; it is easy to visualize and to compute. This way, one less concept seems new and confusing. The function also works for easy visualization because each whole number unit on the graph from x = 0 to x = 10 makes for a slice where we will place a square cross-section. When viewing each of the 10 layers on the 3D model, it's easy to see each step as a piece of the graph without having to think too hard about fractions.
How to compute the volume of this solid
In general, the volume V of a solid of known cross-section oriented along the x-axis with cross-sectional area A(x) from x = a to x = b is \[V = \int_{a}^{b} A(x) dx \] Because our cross-sections are squares, \(A(x) = s^2 = (\sqrt{100-x^2})^2 = 100 - x^2\) (wow, that looks easy to integrate!) Now, calculate the volume by taking the following integral: \[V = \int_{0}^{10} (100-x^2) dx = \frac{2000}{3}\]
How to approximate the volume of this solid
To approximate the volume of this solid, simply take the sum of the volumes of each cross-section. For rectangular prisms like the ones we see making up our cross-sections in the 3D model, the volume is simply length x width x height. Notice that the height of each of our prisms is 1! All we have to do is use right-hand endpoints to compute the side lengths of each cross-section and multiply the length x height. (Because length = height, we can just square the side length of each cross-section to find its volume.) This can be seen in the table below.
This approximated volume is very close to the actual volume! :)
Solids of known cross-section are not as hard to think about as they seem. Starting with the graph of a two-dimensional function, one can slice it into pieces; for each piece, one can take a known shape in three dimensions and visualize it in place of each slice. Often, it is easier to see than to explain. Consider the example below:
For simplicity's sake, we are going to consider the solid bounded by quadrant I and the function \(y = \sqrt{100-x^2}\) seen below where every cross-section perpendicular to the x-axis is a square.
Why I chose this solid
This solid works well for beginners because the first area formula we are taught growing up is the area of a square; it is easy to visualize and to compute. This way, one less concept seems new and confusing. The function also works for easy visualization because each whole number unit on the graph from x = 0 to x = 10 makes for a slice where we will place a square cross-section. When viewing each of the 10 layers on the 3D model, it's easy to see each step as a piece of the graph without having to think too hard about fractions.
How to compute the volume of this solid
In general, the volume V of a solid of known cross-section oriented along the x-axis with cross-sectional area A(x) from x = a to x = b is \[V = \int_{a}^{b} A(x) dx \] Because our cross-sections are squares, \(A(x) = s^2 = (\sqrt{100-x^2})^2 = 100 - x^2\) (wow, that looks easy to integrate!) Now, calculate the volume by taking the following integral: \[V = \int_{0}^{10} (100-x^2) dx = \frac{2000}{3}\]
How to approximate the volume of this solid
To approximate the volume of this solid, simply take the sum of the volumes of each cross-section. For rectangular prisms like the ones we see making up our cross-sections in the 3D model, the volume is simply length x width x height. Notice that the height of each of our prisms is 1! All we have to do is use right-hand endpoints to compute the side lengths of each cross-section and multiply the length x height. (Because length = height, we can just square the side length of each cross-section to find its volume.) This can be seen in the table below.
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