Approximating a Solid with Known Cross Sections

For a solid with known cross section, we can multiply the depth and the area of each cross section, and adds them together to find the total volume of the solid.

We investigate a solid whose base is bounded by the functions of y = sin(x) and y = cos(x), 0.25 π ≤ x ≤ 1.25 π, the cross sections perpendicular to the x-axis are equilateral triangles. The reason of choose this solid is that is a relatively complex object but is generated by two simple equations that we are familiar with. We can suppose this solid is a model of an island.

It is feasible to use the volume by cross section method to calculate total volume of the solid.

We can calculate the side length of each equilateral triangle shaped slice, and then calculate the area of each equilateral triangle. Further, we compute the volume of each slice by multiplying the area and the depth of the slice. By summing up volumes of all the slices, we get the total volume of the solid. The detailed calculation process is shown in the following table:

From the table above, we got the approximation of the volume is 1.358 cubic inches. We can use the following integral to calculate the actual volume of the solid. \[V=\frac{\sqrt{3}}{4} \times ∫_{\pi/4}^{5\pi/4}{(sin⁡(x)-cos⁡(x))^2}dx\] \[=\frac{\sqrt{3}}{4}\pi\] \[=1.360\] The actual volume of the solid is 1.360 cubic inches. The difference between the approximated volume and the accurate volume is 1.360-1.358=0.002 cubic inches. The relative error is 0.002/1.360 ≈ 0.15%.