Introduction
A region in the plane may have a curved or slanted boundary, so it need not be a rectangle. One way to describe such a region is to examine it one vertical line at a time. At each chosen \(x\)-value, the line cuts the region in a segment running from a lower boundary to an upper boundary.
This description is useful because it replaces a two-dimensional shape with a family of one-dimensional intervals. The key is to identify the range of \(x\)-values and, for each one, the bottom and top of the vertical segment.
Vertical Slices and Type I Regions
Fix an \(x\)-value and imagine drawing the vertical line \(x=c\). The part of this line inside a region is called its vertical slice at \(x=c\). If that slice is a single segment, it has a lowest \(y\)-value and a highest \(y\)-value. As \(c\) varies, those endpoint values may trace out curves.
For each fixed \(x=c\) in the interval \([a,b]\), the vertical slice is therefore the segment from \((c,g(c))\) to \((c,h(c))\). The interval \([a,b]\) is the range of \(x\)-values covered by the region. It is not a lower or upper \(y\)-bound: those are the values of \(g(x)\) and \(h(x)\).
When \(g(c)=h(c)\), the slice has zero length: its lower and upper endpoints coincide. This can happen at an endpoint of the \(x\)-interval, as it does at the tip of a triangular region. The sides at \(x=a\) and \(x=b\), when present, are vertical segments; they are not functions giving the lower or upper \(y\)-values across the entire region.
Why the Description Matches the Region
The set notation in the definition has a direct geometric meaning. The conditions \(a\le x\le b\) select which vertical lines to consider. For any selected line \(x=c\), the conditions \(g(c)\le y\le h(c)\) select exactly the points on that line between the two boundary values.
Conversely, a point on the vertical segment from \(y=g(c)\) to \(y=h(c)\) satisfies \(g(c)\le y\le h(c)\), and its \(x\)-coordinate satisfies \(a\le c\le b\). It therefore belongs to the set in the definition. Thus, collecting all these vertical segments for \(a\le c\le b\) gives exactly \(D\), with no points omitted or added.
Worked Examples
Worked Example: The Region Between a Parabola and a Horizontal Line
Consider the region bounded by \(y=x^2\) and \(y=2\). Find its Type I description.
First find where the boundary curves meet by setting their \(y\)-values equal:
These intersection points determine the endpoints of the \(x\)-interval. Between them, the parabola lies below the horizontal line: indeed, \(x^2\le 2\) for \(-\sqrt{2}\le x\le\sqrt{2}\). Thus the lower boundary is \(g(x)=x^2\), and the upper boundary is \(h(x)=2\).
At \(x=0\), the slice runs from \(y=0^2=0\) to \(y=2\). At either endpoint, \(x^2=2\), so the slice has zero length. The resulting description is:
Checking the bounds at \(x=0\) gives \(0\le y\le2\), as expected. At \(x=\sqrt{2}\), both \(y\)-bounds equal \(2\), confirming that the slice closes at the intersection.
Worked Example: A Triangle Between Two Lines
Consider the triangle bounded by \(x=0\), \(y=x\), and \(y=4-x\). Describe it using vertical slices.
The two sloping sides meet where their \(y\)-values agree:
The triangle extends from the vertical side \(x=0\) to the meeting point at \(x=2\). For \(0\le x\le2\), the lower line is \(y=x\) and the upper line is \(y=4-x\). To check their order, subtract the lower value from the upper value:
At \(x=0\), the slice runs from \(y=0\) to \(y=4\). At \(x=1\), it runs from \(y=1\) to \(y=3\). At \(x=2\), both lines give \(y=2\). Therefore, the Type I description is:
The endpoint slices check the geometry: the first is the vertical side of the triangle, and the last has collapsed to its top vertex.
Worked Example: The Region Under a Square-Root Curve
Consider the region above the \(x\)-axis, below \(y=\sqrt{x}\), and between \(x=0\) and \(x=4\). Identify its vertical-slice description.
The specified \(x\)-interval is \([0,4]\). At each \(x\) in this interval, the \(x\)-axis gives the lower value \(y=0\), while the curve gives the upper value \(y=\sqrt{x}\). The square root is defined here because \(x\ge0\), and it is nonnegative, so \(0\le\sqrt{x}\) throughout the interval.
For instance, at \(x=1\), the slice runs from \(y=0\) to \(y=\sqrt{1}=1\). At \(x=4\), it runs from \(y=0\) to \(y=\sqrt{4}=2\). At \(x=0\), both bounds equal zero. The full description is:
These endpoint and interior checks confirm that the lower boundary stays on the \(x\)-axis while the upper boundary rises from \(0\) to \(2\).
When One Pair of Boundary Functions Is Not Enough
The definition requires each vertical slice, for every \(x\) in the chosen interval, to be a single segment. Some regions do not have this property. For example, a region shaped like a ring has vertical lines that pass through the region in two separate segments: one above the hole and one below it. A single pair of lower and upper functions would include the points in the hole between those segments, so it would describe a larger set than the ring.
This limitation does not mean that such a region can never be studied using vertical slices. It means only that one Type I description, with one interval of \(y\)-values for each \(x\), does not describe it exactly. A region with multiple vertical pieces may need to be divided into simpler parts. Recognizing whether a slice is connected is therefore part of checking that a Type I description fits.
Common Mistake: Confusing the \(x\)-Range with the \(y\)-Bounds
In the triangular example, a tempting but incorrect description is to let \(x\) range from \(0\) to \(4\), because the sloping lines meet the vertical axis at \(y=0\) and \(y=4\). But those are \(y\)-values on the line \(x=0\), not endpoints of the triangle’s \(x\)-range.
The incorrect choice also makes the proposed lower and upper functions switch order: at \(x=3\), they give \(y=3\) and \(y=4-3=1\), respectively. Since \(3>1\), those bounds do not describe a vertical segment running from the proposed lower value to the proposed upper value. The lines meet at \(x=2\), and the triangle ends there. The correct \(x\)-range is \(0\le x\le2\), on which \(x\le4-x\).
Why Vertical Slices Matter
Vertical slices give a systematic way to organize a region that is not rectangular. Instead of treating its boundary as one complicated outline, we describe the region through simpler segments and track how their endpoints change with \(x\). The functions \(g\) and \(h\) record that change.
This geometric description is the foundation for expressing quantities over general planar regions using iterated integrals. Before setting up such an integral, it is essential to know what the region actually contains: which \(x\)-values occur, and where each vertical slice begins and ends. A correct slice description keeps the geometry and the eventual calculation aligned.
Check Your Understanding
For each question, focus on the range of \(x\)-values and the two endpoints of a vertical slice.
- In a Type I description, what does the interval \([a,b]\) specify, and what do the functions \(g\) and \(h\) specify?
- For the region between \(y=x^2\) and \(y=5\), what equations determine the endpoints of the \(x\)-interval?
- For the triangle bounded by \(x=0\), \(y=2x\), and \(y=6-x\), which curve is lower on the region’s interior?
- Why does a ring-shaped region fail to have one connected vertical slice at every \(x\)-value?
- What condition must the lower and upper boundary functions satisfy throughout the \(x\)-interval?