Calculus I · Unit 3B · lesson

Choosing Vertical or Horizontal Slices

Concept

Learning objectives

Choose the orientation that avoids unnecessary piecewise integrals.

Choosing Vertical or Horizontal Slices

Explanation

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 $y$ , especially when the boundaries are already given as $x=g(y)$ . 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 $dx$ , horizontal slices have thickness $dy$ . 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.

Worked example

A sideways region

The region between $x=y^2$ and $x=2y+3$ is naturally described with horizontal slices. Intersections solve $y^2=2y+3$ , giving $y=-1,3$ . Since the line lies to the right,

A=\int_{-1}^{3}[(2y+3)-y^2]dy.

A vertical setup would require solving for $y$ and splitting the region.

Interactive checku3b-slice-choice-01

For the region bounded by $x=y^2$ and $x=y+2$ , which slice orientation gives one integral most naturally?

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Show hint

Both boundaries are already functions of $y$ .

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After the explanation

Use the section idea

Reading lens

Let the axis and slice orientation determine every distance; top-minus-bottom, right-minus-left, radii, and shell height must all come from the same picture.

Mental model

Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.

Decision

Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.

Common trap

Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.

Check yourself

Do the slice dimensions remain nonnegative on the full interval, and do their units multiply to area or volume?

Source & rights

Original instruction with traceable references.

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