Calculus I · Unit 3B · lesson
Choosing Vertical or Horizontal Slices
Learn Choosing Vertical or Horizontal Slices through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Area and volumeWhat this section is building
Learn Choosing Vertical or Horizontal Slices through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.
Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.
Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.
Learning objectives
Choose the orientation that avoids unnecessary piecewise integrals.
Choosing Vertical or Horizontal Slices
Choose the orientation that describes the region simply
Vertical slices are not automatically superior. Some regions are naturally described by right-minus-left as functions of , especially when the boundaries are already given as . The orientation should be chosen to make each slice a single segment with a simple length and to avoid unnecessary piecewise formulas.
Draw both a vertical and a horizontal test slice before committing to an integral. Ask which boundaries the slice touches and whether those boundaries change as the slice moves. The differential must match the orientation: vertical slices have thickness , horizontal slices have thickness . This small planning step often turns a two-integral mess into one clean calculation.
A region may be simple vertically but piecewise horizontally, or vice versa. Sketch first. Draw one representative slice. Label its length using the axis perpendicular to the slice.
A sideways region
The region between and is naturally described with horizontal slices. Intersections solve , giving . Since the line lies to the right,
A vertical setup would require solving for and splitting the region.
u3b-slice-choice-01For the region bounded by and , which slice orientation gives one integral most naturally?
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Both boundaries are already functions of .
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