Choosing dx or dy for a Volume Integral

Every volume-by-slicing problem needs a direction to cut in. Slice perpendicular to the x-axis and you integrate dx, with each slice's radius or height written in terms of x. Slice perpendicular to the y-axis and you integrate dy, with everything rewritten in terms of y. The two setups describe the exact same solid, but one of them usually turns into a clean integral and the other turns into a mess of square roots or a function that isn't even single-valued.

The decision rule is about the boundary, not the axis. Look at how the region is described. If the curves are naturally functions of x (like y = x²), and the axis you're spinning around is the y-axis, slicing in y means solving for x in terms of y, which lines up a disc's radius directly with the rotation axis. If instead the outer edge of the region is a simple horizontal line and the inner edge is the curve, slicing in x gives you a clean washer even when rotating about the x-axis. The direction that matches the axis of rotation is common, but it's a consequence of convenience, not a rule you apply blindly.

Sometimes neither horizontal nor vertical disc slicing works cleanly, because one boundary isn't a single function of the slicing variable. That's when the shell method steps in: you slice parallel to the axis of rotation instead of perpendicular to it, using radius and height instead of area times thickness. This is the same judgment call you made choosing dx or dy for area between curves — read which variable makes each boundary a clean, single function, and integrate in that direction.