What is the difference between the methods for calculating volume in calculus like disk,washer,cross section, shell, etc?

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What is the difference between the methods for calculating volume in calculus like disk,washer,cross section, shell, etc?

In: Mathematics

The method is the same. You basically take the function that describes the outer and inner surfaces of the shape (or outer only, if it’s a solid shape with no voids, ie. cylinder versus tube). If the axis runs through the center of your shape you can integrate half and double it. If the axis is outside your shape, or doesn’t run through the center, you’ll have to integrate multiple curves and sum/subtract to get the volume of the shape.

Set up the shape, then draw a narrow slice or partition of width dx (or dy, or dz depending on shape/orientation). The approach is integrate the outer surface function, then integrate the inner surface function and subtract it from the outer surface. You can combine those into a single equation and integrate once.

The goal is set up the equation such that your incremental partition width can cross the whole shape from one side to the other, and each function is has the same independent variable.

All volume formulas are doing the same thing…slice the volume up into very tiny pieces that have a “simple” formula for their volume, then add them all up.

The difference between the different methods is what small shape you’re using. The choice is arbitrary to the end result but how easy it is to do the math depends on matching the small volumes to the big one. Hole in the middle? Washers. Cylindrical? Disks. Spherically symmetric? Shells. Arbitrary? Cross sections (hopefully sliced in a direction that gives you nice cross sections). They’re all about choosing the setup to make the intermediate steps simpler so the path to the end is easier.

Edit: it’s entirely possible to derive the correct formula for the volume of a sphere using small cubes…it just sucks. The geometry is horrible. If you use tiny shells it’s really easy.