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Solids of Known Cross-Section

While integration is often defined as a way to compute areas, a classic application of it is to compute other quantities including arc lengths and volumes. Cavalieri’s Principle implies that volumes can be computed by essentially integrating cross-sectional areas.

For example, if a cylinder of radius $r$ and height $h$ is laying lengthwise along the $x$-axis, then its cross-sections at any value of $x$ are circles of radius $r$ and thus area $\pi r^2$. Therefore, its volume can be computed as $$\int_0^h \pi r^2\,\mathrm{d}x=\pi r^2 x|_0^h=\pi r^2h.$$

A slightly harder problem is to find the volume of a sphere of radius $r$. If this sphere is centered at the origin, then the cross-sections at any value of $x$ are circles whose radiuses, by the Pythagorean Theorem, are $\sqrt{r^2-x^2}$. The areas of these cross-sections, then, are $\pi (\sqrt{r^2-x^2})^2$, so the volume of the sphere is $$\int_{-r}^r \pi (r^2-x^2)\,\mathrm{d}x=\pi\left(r^2x-\frac{x^3}{3}\right)\Big|_{-r}^r=\frac{2\pi r^3}{3}-\frac{-2\pi r^3}{3}=\frac{4\pi r^3}{3}.$$

The upshot is that volumes of a solid can be computed so long as the areas of its cross-sections are understood. The washer method for volumes of revolution is a special case of this idea. Students often have trouble with these problems, though, citing an inability to picture the solid. I always emphasize that one does not need to understand the 3-dimensional picture to compute the volume, just the 2-dimensional pieces, but that only comforts students so much.

So, this semester I assigned my students each a solid described by its cross-sections and had them construct a 3-dimensional scale model of the solid. I had had the idea for the assignment before, but doubted that students would be able to construct the models well enough for it to be both beneficial and not too frustrating. I was encouraged, though, by seeing accounts of other teachers’ successes with it, and gave it a try. I was so incredibly impressed with my students’ work that I decided to document it in this blog.

Solid 1: The base of the solid is bounded by the parabola $y=x^2/2$ and $y=2$ and the cross-sections perpendicular to the $y$-axis are semicircles.

Parabolic Base with Semicircular Cross-Sections

Parabolic Base with Semicircular Cross-Sections
Parabolic Base with Semicircular Cross-Sections

The last of these wins the award for most creative medium. It is made from college mail, whose sturdiness and expendability make it perfect for student construction projects.

Solid 2: The base of the solid is the region bounded by the parabola $y^2=3x$ and the line $x=3$ and cross-sections perpendicular to the $x$-axis are equilateral triangles.

Parabolic Base with Equilateral Triangle Cross-Sections

Solid 3: The base of the solid is the circle $x^2+y^2=4$ and cross-sections perpendicular to the $x$-axis are squares.Circular Base with Square Cross-Sections

I particularly liked the braces at the top of the cross-sections, which added structural support.

Solid 4: The base of the solid is the region bounded by $y=1/x$, $y=0$, $x=1$, and $x=4$ and the cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuses along the bases.

Hyperbolic Base with Isosceles Right Triangle Cross-Sections

Clearly, my worries that students would not be able to handle the project were completely unfounded. In fact, I was thanked by some students for giving this assignment because it helped the students understand solids and volumes better. I plan to add this project to my regular line-up, and recommend it to anyone else teaching integral calculus.

One Comment

  1. This looks fantastic — and given how intellectually challenging it is to picture these solids, it’s got to be a huge help to the kids!

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