IB Maths AI HLFurther IntegrationPaper 1 & 2~6 min read
Area Between a Curve and a Line
The area trapped between two graphs is the integral of top − bottom across the region. Find where they intersect (those are your limits), decide which graph is on top, and integrate the difference. Whether the boundary is a line or another curve, the method is identical.
📘 What you need to know
Top minus bottom: area = ∫ab (ytop − ybottom) dx.
Limits = intersections: solve the two equations equal to each other to find a and b.
Identify the upper graph on the interval — test a point if unsure.
Difference handles the axis: subtracting means you don’t worry separately about below-axis parts within one region.
Multiple regions: if the graphs cross more than twice, split and add the magnitudes.
GDC: evaluate the integral directly on Paper 2 to check.
Top minus bottom
Area between two graphsA = ∫ab (ytop − ybottom) dx✓ area between curves is in the formula booklet
The enclosed region
The region is bounded above by one graph, below by the other, between their intersection points a and b.
🤔 Why does subtracting work even below the axis?
The height of the region at any x is the gap between the two graphs: ytop − ybottom. That gap stays positive everywhere in the region, regardless of whether the graphs are above or below the x-axis. So a single integral of the difference gives the area directly — no separate sign-handling needed within one enclosed region.
🧠 “Intersect, identify, integrate the difference”
Intersect the graphs for the limits, identify which is on top, then integrate (top − bottom). If you get a negative, you subtracted the wrong way round — swap them.
Carrying it out
🧭 Recipe — area between a curve and a line
Find intersections: set the two equations equal and solve for x — these are a and b.
Decide top vs bottom: test an x between them, or sketch.
Integrate∫ab (ytop − ybottom) dx.
Evaluate top minus bottom.
More than two crossings? Split at each, add magnitudes.
Quick top/bottom test: pick any x inside the region and put it into both equations — whichever gives the larger y is the top graph.
Worked examples
WE 1
Find the area enclosed between y = x + 2 and y = x2
Intersect, then the line is above the parabola between the roots.
x² = x + 2 → x² − x − 2 = 0 → x = −1, 2∫−12 (x + 2 − x²) dx = [x²2 + 2x − x³3]−12= 103 − (−76)area = 92 square units
WE 2
Find the area enclosed between y = x2 and the line y = 4
The horizontal line is on top; intersections at x = ±2.
Find the total area enclosed between y = x3 and y = x
They cross three times, giving two regions — and the top graph swaps. Split and add magnitudes.
x³ = x → x = −1, 0, 1on (0,1): x on top → ∫01 (x − x³) dx = 14on (−1,0): x³ on top → area = 14 (symmetry)total = 14 + 14area = 12 square unitstraight ∫−11 = 0 — must split
WE 5
Find the area enclosed between y = √x and the line y = x
Between 0 and 1, √x lies above the line.
√x = x → x = 0, 1∫01 (√x − x) dx = [23 x3/2 − x²2]01= 23 − 12area = 16 square unit
💡 Top tips
Solve for intersections first — they’re your limits.
Test a point inside to confirm which graph is on top.
Negative answer? You subtracted bottom − top — just take the modulus.
Three crossings = two regions: split and add magnitudes.
Sketch to see the enclosed shape clearly.
GDC evaluates ∫(top − bottom) directly on Paper 2.
⚠ Common mistakes
Subtracting the wrong way — bottom − top gives a negative.
Integrating each graph separately and mishandling below-axis parts, instead of the single difference.
Missing a third intersection, so two regions get merged and cancel.
Using wrong limits — they must be the intersection x-values.
Forgetting to split when the top graph changes.
That wraps up Further Integration. The unit grew from one root idea — the definite integral as a signed area — into a full toolkit: standard integrals for special functions, the reverse chain rule and substitution for composites, then a sequence of area techniques — definite integrals and their properties, negative integrals below the axis, area against the y-axis, and finally the area between two graphs as the integral of top − bottom. Every one comes back to the same move: integrate a difference between limits, and mind the sign.
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