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.
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