Estimating Material for 3D Printed Objects

You work at a public library’s makerspace, helping visitors estimate how much filament their 3D prints will require. Today, a student needs help calculating material usage for a custom phone stand they designed. The object has a complex shape that can be described using mathematical surfaces, and the print settings create varying infill density throughout different regions of the object. Your task is to use triple integrals to determine the total volume, account for the density variation, and calculate the average density to help the student estimate their filament cost. The library charges based on material weight, so accurate calculations will help the student budget appropriately.