Volumes Using Known Cross-Sections

In the last few lessons, we have looked at cross sections that were in the shape of disks.
Now, we will look at cross-sections that are other shapes.

Problem:

A solid has a circular base of radius 4 units. Find the volume of the solid if every plane
section perpendicular to a fixed diameter is an equilateral triangle.

Solution:

Draw a circle on a three dimensional coordinate system, using the x-axis as the diameter
of the circle. The circle will intersect the x-axis and the y-axis at -4 and 4. The equation
of the circle is x² + y² = 16. Draw a representative cross section as an equilateral triangle
parallel to the z-axis. This cross-section has sides equal to 2y and area A = \/3 y².
(The height of the triangle is \/3 y, using the Pythagorean Theorem.) Since y = \/(16 - x²),
then A = (\/3) (16 - x²). The differential (or width of the triangular prism) is dx.

cubic units