Below the axis โ the integral is negative: Where a curve is below the x-axis, the definite integral is negative.
The area is the magnitude: take the absolute value (or split at the x-intercepts and add the magnitudes).
IB-style question โ area below the axis
Find the area between y = xยฒ โ 4 and the x-axis from x = 0 to x = 2.
Step by step
- Integrate (the curve is below the axis here).
- Area is the magnitude.
Final answer
Area = 16/3 (the integral is โ16/3 because the region is below the axis).
Area is never negative: A negative integral just means the region is below the axis โ report the positive area.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience โ and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Integrate (top โ bottom): The area between two curves from x = a to x = b is โซโแต (top โ bottom) dx, where 'top' is the upper curve.
This works even below the x-axis, because you subtract the lower curve.
The region between two curves โ its area is the integral of (upper curve โ lower curve) over the interval.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
IB-style question โ between two curves
Find the area enclosed between y = x and y = xยฒ (from x = 0 to x = 1).
Step by step
- On [0,1], y = x is above y = xยฒ. Integrate the difference.
- Evaluate.
Final answer
Area = 1/6.
Decide which curve is on top: Test a value between the limits (or sketch) to see which curve is higher โ that's the 'top'.
Study smarter, not longer
Most students waste 40% of study time on topics they already know. Our AI tracks your progress and optimizes every minute.
Limits come from intersections / intercepts: If the region is enclosed, the limits are where the curves meet โ solve top = bottom (or, for a curve and the axis, solve f(x) = 0).
Then integrate (top โ bottom) between those x-values.
IB-style question โ find limits then area
Find the area enclosed between y = 6x โ xยฒ and y = x.
Step by step
- Intersections: 6x โ xยฒ = x โ 5x โ xยฒ = 0 โ x(5 โ x) = 0 โ x = 0, 5.
- Top is 6x โ xยฒ; integrate the difference 5x โ xยฒ over [0,5].
Final answer
Area = 125/6 โ 20.8.
Solve top = bottom first: Always find the intersection x-values before integrating an enclosed region โ they are your limits.