AP® Calculus AB · Unit 8 · Topic 8.5
Finding the Area Between Curves Expressed as Functions of y
When a region is bounded by sideways-opening curves, slice it horizontally instead of vertically: integrate (right x minus left x) with respect to y over the y-values where the curves meet.
Area between curves always comes from stacking thin strips and adding them up. Usually you slice vertically: strips of height (top minus bottom), integrated in x. But some regions open sideways — bounded by a curve like x = y2, which fails the vertical line test and can't be written as a single function of x without splitting the region into pieces. Turn the slices on their side. Now each strip has width (right minus left), stretches a tiny bit in the y direction, and you integrate with respect to y instead.
The mechanism mirrors the vertical case exactly, just swapped. Find where the curves meet by setting their x-expressions equal and solving for y — these become your limits of integration. Then integrate (x_right(y) - x_left(y)) dy between those y-values. Take x = y2 and x = y + 2: setting y2 = y + 2 gives y2 - y - 2 = 0, so y = -1 and y = 2. On that interval the line sits to the right of the parabola, so the area is the integral from -1 to 2 of ((y+2) - y2) dy.
The real skill is deciding which way to slice before you integrate anything. Look at the picture: if vertical lines cut the region in one continuous piece, slice in x; if horizontal lines do, slice in y. This same judgment call resurfaces when you build solids of revolution — choosing to rotate around the x-axis or y-axis, and whether you get discs or washers, depends on exactly this same read of which direction slices the region cleanly.
Key terms
- Horizontal slice
- A thin rectangular strip of width (right x minus left x) and height dy, used to build area when integrating in y.
- Right minus left
- The setup for horizontal slices: subtract the x-value of the left boundary from the x-value of the right boundary.
- Vertical line test
- A check for whether a curve is a function of x; if a vertical line hits it more than once, it fails and x = y2-style slicing in y is needed.
- Limits of integration in y
- The y-values where the two boundary curves intersect, found by setting their x-expressions equal and solving for y.
Watch it explained
- How it worksHow do horizontal slices of width right minus left set up an integral in y?
- Worked exampleFind the area between x = y2 and x = y + 2 by integrating with respect to y.
- Common mistakeWhy isn't integrating in x always possible for a sideways-opening region?
- Going furtherWhat tells you at a glance whether to slice vertically or horizontally?
- Ties togetherHow does choosing to slice horizontally here reappear when you choose between discs and washers later in the unit?
Check yourself
End of topic.