Let R be the region enclosed by the x-axis, the graph y = x2, and the line x = 4. Write an integral expression for the volume of the solid whose base is R
whose slices perpendicular to the y-axis are equilateral triangles.
The region R is the same as in the example:
We're slicing perpendicular to the y-axis again. The base of a slice still has length , but now the slice is an equilateral triangle instead of a square. Our volume looks like a Toblerone.
The area of an equilateral triangle with side-length s is
so the volume of a slice is
The variable y goes from 0 to 16 in the base region R, so the volume of the solid is