This is where the integral grows up. You already know it measures area under a curve —
now you'll measure the area trapped between two. Master the setup here and every volume in this unit
becomes the same move in disguise.
1 · The whole idea: top minus bottom
Forget formulas for a second. If one curve sits above another, the area trapped between them
is just the area under the top curve minus the area under the bottom curve. Two areas you already know how
to find, subtracted. That's the entire concept — everything else is bookkeeping.
Here's why it collapses into a single clean integral. Picture slicing the region into many thin
vertical strips. Each strip is a skinny rectangle: its width is a tiny dx, and
its height is the distance from the bottom curve up to the top curve — that's
(top − bottom). The area of one strip is height × width,
(top − bottom)·dx. Add up every strip across the interval and you have the integral.
(You met this "limit of a sum of strips" idea back in Unit 6 — here each term in the
sum is simply a top-minus-bottom strip.)
top(x)
The upper curve — whichever function is higher on the interval. Its area would be ∫ top dx alone.
− bottom(x)
Subtract the lower curve. Removing the area beneath the bottom curve leaves exactly the strip between them.
[top − bottom]
The height of one strip at position x — the green value minus the amber value. Always upper minus lower, so the height stays positive.
Sum the strips from a to b (the intersection points). That total is the area between the curves.
The #1 mistake: guessing which curve is on top instead of checking. One test
point between the intersections settles it every time.
Take this page's curves at x = 0:
the line y = x + 2 gives 2, the parabola
y = x² gives 0. Since 2 > 0,
the line wins — integrate (x + 2) − x², in that order.
Subtract the other way and you'd get a negative area — the tell-tale sign you
flipped them. Area is always (upper − lower).
One more piece completes the setup: where do the limits a and
b come from? They're the x-values where the curves cross — found by setting the
two functions equal and solving. The visual below works a full example start to finish, bounds and all.
2 · A complete example, start to finish
Let's work the whole thing for y = x + 2 and
y = x² — every step you'd actually do, in order. Click through:
find the bounds, decide which curve is on top, then subtract the two areas.
That's the entire method, every time: (1) set the curves equal to find where they cross —
those are your bounds; (2) test a point to see which is on top; (3) integrate top minus bottom. The
area between isn't a new mystery shape — it's the area under the top with the area under the bottom carved away,
here 15/2 − 3 = 9/2.
3 · When vertical strips fail: integrate in y
Vertical strips assume a single "top" and a single "bottom" the whole way across. But some
regions don't cooperate — the boundary on top or bottom changes partway, which would force you to split
into multiple integrals. Often there's a slicker fix: turn the strips sideways.
If you slice with horizontal strips instead, each strip's width runs from a
left curve to a right curve, and you integrate in
y. The formula is the same idea rotated 90°: (right − left)
instead of (top − bottom), summed over y.
This is the region bounded by x = y + 2
and x = y² — the same shape as before, lying on its side.
Drag the strip up through the region: each horizontal strip runs cleanly from the parabola on the
left to the line on the right, with width
(right − left).
Area —
same answer, because it's the same region. The art is choosing the strip direction that needs the
fewest integrals.
Sweep the strip y = 0.50
Strip width = right − left = (y+2) − y²
–
sum these widths over y → area 9/2
✗ Vertical strips (in x)
The bottom boundary switches from the parabola to the line at x = 1, so the top-minus-bottom rule changes partway. You'd need to break it into two integrals:
✓ Horizontal strips (in y)
Every strip runs from the parabola (left) to the line (right) the whole way up — one rule, no switch. Just one integral:
Each horizontal strip runs from the parabola (left) to the line (right). Its width is widest in the middle and shrinks to zero at the corners (y = −1 and y = 2), where the curves meet. Add up every strip's width across y and you get the area, 9/2 — same as the vertical-strip answer.
How to choose: look at the region. If a single function is clearly on top and
another clearly on the bottom across the whole interval → vertical strips, integrate in x.
If instead a single function is clearly on the left and another on the right → horizontal strips, integrate in
y. Pick whichever gives one clean integral instead of several.
4 · When curves cross: split the integral
What if the two curves cross inside your interval? Then "top" and "bottom"
swap places at the crossing — the curve that was higher is suddenly lower. A single
(top − bottom) integral can't handle that, and it'll quietly give you the wrong answer.
Here are y = x and
y = x³ on [−1, 1]. They cross
at x = 0. On the left half the cubic is on top; on the right half the
line is on top. The roles flip.
Left piece [−1, 0]
–
Right piece [0, 1]
–
Total
–
The dangerous part: if you ignore the crossing and just compute
, you get exactly 0 — the two halves cancel, because one comes
out negative. Zero area for an obviously non-empty region is the classic tell that you forgot to split. The fix is
always the same: find every intersection inside the interval, split there, and make each piece (upper − lower)
so all areas count as positive. (Same logic as splitting at sign changes for distance, last page.)
Three situations, one skill. Top − bottom sets up every area. dx vs dy is just
choosing the strip direction that's cleanest. Splitting handles curves that trade places. Underneath, it's
always the same move: sum up (upper − lower) strips, set up so every piece is positive.
5 · Drill it
Six questions across setting up top − bottom, choosing dx vs dy, and splitting at
crossings. Each gives a full worked solution after you answer.
Score: 0 / 0
This setup skill is the foundation for everything next: spin a region around an axis and the same
(top − bottom) thinking becomes volume, or stack known shapes on it for
cross-sections. Back to Unit 8.