Calculus I › Integrals › full formula sheet
Area between curves
Area is always top minus bottom — find where the curves cross first, then subtract in the right order.
Notation on this page: top(x) is the upper curve, bottom(x) the lower one; a, b are usually their intersection points.
Before this lesson: FTC Part 2 · Splitting intervals
Where it comes from
The problem: ∫ab f(x) dx measures area between one curve and the x-axis. But most regions are trapped between two curves — neither of which is the axis. The fix: at each x, the region’s vertical slice has height = top(x) − bottom(x). Integrate the heights.
The tempting blind subtraction:
Before reading on: on [1,2], is x or x2 on top? Plug in x = 1.5. Before reading: what sign does the wrong-order integral ∫12 (x−x2) dx give — and why is that impossible for a geometric area?
On [1, 2], x² is above x — so x − x² is negative, and the integral gives [x²/2 − x³/3]12 = (2 − 8/3) − (1/2 − 1/3) = −5/6: a negative “area.” The subtraction order is not cosmetic — it decides the sign:
Intuition: “area under the top curve” minus “area under the bottom curve” leaves exactly the strip between them — the parts below the bottom curve cancel out of both.
Derivation
Slice the region into thin vertical strips, add them up, take the limit. Each strip is a rectangle whose height is the vertical gap between the curves.
Equivalently: A = ∫ab |f(x) − g(x)| dx. The top−bottom form is just |f−g| with the crossings split out — on each piece you know which curve is on top, so the absolute value is unnecessary.
How to use it
Before reading on: y = x3 and y = x cross three times on [−1, 1], so top and bottom trade places mid-interval. Before reading: can one integral still give the area — and if not, what must you do first?
The procedure, every time:
- Find the intersections: solve f(x) = g(x). These are usually your limits a and b — the region’s left and right edges.
- Decide top vs bottom: pick a test point between the intersections and evaluate both functions. The larger value is on top there.
- Write ∫ab (top − bottom) dx and integrate.
- If the curves cross inside [a, b], split there. Top and bottom swap at every crossing — one blind integral lets signed areas cancel.
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the basic move: y = x and y = x² on [0, 1]
- Intersections: x = x² ⇒ x = 0 or x = 1. (Why solve? The enclosed region starts and ends where the curves meet.)
- Top vs bottom: test x = 1/2: y = x gives 1/2, y = x² gives 1/4. So top = x, bottom = x².
- Integrate: ∫01 (x − x²) dx = [x²/2 − x³/3]01 = 1/2 − 1/3 = 1/6.
- Sanity check: the region sits inside the unit square, thin near both ends — 1/6 of the square is plausible. ✓
Your turn: Find the area between y = x2 and y = x3 on [0, 1].
Answer: 1/12
Intersections: x2 = x3 ⇒ x = 0, 1. Test x = 1/2: 1/4 > 1/8, so top = x2. Integrate: ∫01 (x2−x3) dx = [x3/3 − x4/4]01 = 1/3 − 1/4 = 1/12. Sanity: a thinner region than the x-vs-x2 one, so 1/12 < 1/6. ✓
Example 2 — against the axis: y = 4 − x² and y = 0
- Intersections: 4 − x² = 0 ⇒ x = ±2.
- Top vs bottom: the parabola arches above the x-axis between −2 and 2 — top = 4−x², bottom = 0.
- Integrate: ∫−22 (4−x²) dx = [4x − x³/3]−22 = (8 − 8/3) − (−8 + 8/3) = 16 − 16/3 = 32/3.
- Sanity check: the region fits in a 4×4 box (area 16); 32/3 ≈ 10.67 < 16, and it’s most of the box. ✓
Your turn: Find the area between y = 9 − x2 and y = 0.
Answer: 36
Intersections: 9 − x2 = 0 ⇒ x = ±3. The parabola arches above the axis between them. Integrate: ∫−33 (9−x2) dx = [9x − x3/3]−33 = 18 − (−18) = 36. Sanity: fits in a 6×9 box (area 54); 36 < 54, most of the box. ✓
Example 3 — a trig region: y = sin x and y = 0 on [0, π]
- Intersections: sin x = 0 at x = 0 and x = π (the given bounds).
- Top vs bottom: sin x ≥ 0 on [0, π] — top = sin x, bottom = 0.
- Integrate: ∫0π sin x dx = 2 (the area-2 anchor).
Your turn: Find the area between y = cos x and y = 0 on [−π/2, π/2].
Answer: 2
cos x ≥ 0 there, so top = cos x, bottom = 0. Integrate: ∫−π/2π/2 cos x dx = [sin x]−π/2π/2 = 1 − (−1) = 2 — the same area-2 anchor as the sine hump, shifted. ✓
Example 4 — the swap: y = x³ and y = x on [−1, 1]
- Intersections: x³ = x ⇒ x = −1, 0, 1. Three crossings — top and bottom swap at x = 0.
- Test each piece: at x = −1/2: x³ = −1/8 > −1/2 = x — top = x³ on [−1, 0]. At x = 1/2: x = 1/2 > 1/8 = x³ — top = x on [0, 1].
- Split and integrate: ∫−10 (x³−x) dx + ∫01 (x−x³) dx = [x⁴/4 − x²/2]−10 + [x²/2 − x⁴/4]01 = 1/4 + 1/4 = 1/2.
- Sanity check: one blind integral ∫−11 (x³−x) dx = 0 — the lobes cancel. But geometric area can’t cancel; the split is mandatory. ✓
Your turn: Find the area between y = x and y = x2 on [−1, 1].
Answer: 1
Crossings at x = 0 and x = 1: on [−1,0] test x = −1/2 gives top = x2; on [0,1] top = x. Split: ∫−10 (x2−x) dx + ∫01 (x−x2) dx = 5/6 + 1/6 = 1. Sanity: the blind integral ∫−11 (x−x2) dx = −2/3 — signed area lies; the split is mandatory. ✓
Memorization tips
- Say it aloud: “top minus bottom, between the crossings.” Intersections first, subtraction order second.
- The test point: one x-value between intersections settles top vs bottom forever. Five seconds, no sign errors.
- Crossings split: every interior intersection is a mandatory split point. No exceptions — signed area cancels otherwise.
- Sketch first: even a rough doodle shows which curve is on top and where they meet. The 30-second sketch prevents the 5-minute sign error.
- Symmetry halves work: symmetric region about the y-axis → compute the right half and double (Example 2: 2 × 16/3).
- Sanity-cage the answer: the area must be positive and smaller than the bounding box. Negative or enormous → recheck the order.
Final challenge
Five mixed questions — intersections, a √2 answer, and the swap trap. Score 5/5 and areas are yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is the area between two curves?
A = ∫ab [top(x) − bottom(x)] dx: integrate the vertical distance between the curves. Top minus bottom keeps every slice’s height non-negative.
Why must it be top minus bottom?
Because a slice’s height is top(x) − bottom(x) ≥ 0. Integrating bottom − top gives negative “area.” Example: y = x vs y = x² on [1, 2] gives −5/6 if you subtract in the wrong order.
How do I find the limits a and b?
Solve f(x) = g(x) — the intersection points are where the enclosed region starts and ends. If the curves cross inside, split there and do top−bottom on each piece.
What if the curves swap which is on top?
Split at the crossing point. Example: y = x³ vs y = x on [−1, 1] needs ∫−10(x³−x) dx + ∫01(x−x³) dx = 1/4 + 1/4 = 1/2 — one blind integral gives 0.
Is area between curves the same as ∫|f−g|?
Yes — top − bottom is exactly |f − g| when you split at every crossing. The top−bottom form just tells you which order to subtract on each piece.
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