Evaluate definite integrals and areas under curves
Evaluate a definite integral from an antiderivative and interpret signed area when a curve or line lies above or below an axis.
Section 14 · Calculus
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does the definite integral ∫ₐᵇ f(x)dx represent geometrically?
What you will learn
- Evaluate a definite integral by subtracting antiderivative values
- Interpret a definite integral as signed area
- Find the positive area between a line or curve and an axis over a stated interval
Substitute the upper and lower limits into an antiderivative
If F′(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). The arbitrary constant cancels, so it is usually omitted during a definite-integral calculation.
Read the order carefully: upper limit first, then subtract the lower-limit value. Brackets protect negative values when substituting endpoints.
A definite integral is signed area
Where f(x) is above the x-axis, its integral contributes positive area; where it is below, it contributes negative area. A geometric area is non-negative, so split at crossings or reverse the subtraction when needed.
For area between a curve y = f(x) and a line y = g(x), integrate top minus bottom over each interval where their order stays fixed.
Pause and explain
If a curve is below the x-axis throughout an interval, what sign does its definite integral have?
Worked example
Find the area enclosed between y = 2x and y = x² for 0 ≤ x ≤ 2.
Show the worked solution
- On this interval, 2x is above x² because 2x − x² = x(2 − x) ≥ 0.
- Area = ∫₀² (2x − x²) dx = [x² − x³/3]₀².
- Evaluate upper minus lower: 4 − 8/3 = 4/3.
Answer Area = 4/3 square units.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the area between y = x and y = x² for 0 ≤ x ≤ 1.
Give me a hint
On this interval x is above x².
Compare my reasoning
- Area = ∫₀¹ (x − x²) dx.
- An antiderivative is x²/2 − x³/3.
- Evaluate at 1 and subtract the value at 0.
Area = 1/2 − 1/3 = 1/6 square units.
Look for these in your work
- Top curve selected
- Difference integrated
- Limits evaluated in order
Find ∫₀² 3x² dx and interpret it as area.
Give me a hint
Use the antiderivative x³.
Compare my reasoning
- An antiderivative of 3x² is x³.
- Evaluate 2³ − 0³.
- The function is non-negative on the interval, so integral and area agree.
8 square units.
Look for these in your work
- Antiderivative correct
- Upper and lower limits applied
- Signed area interpreted
A water-level graph y = 4 − x is measured from x = 0 to x = 6. Find its signed integral and the total geometric area between it and the x-axis.
Give me a hint
The line crosses the axis at x = 4; split the interval there for geometric area.
Compare my reasoning
- Signed integral is [4x − x²/2]₀⁶ = 24 − 18 = 6.
- Area above the axis from 0 to 4 is ∫₀⁴(4 − x)dx = 8.
- Area below the axis from 4 to 6 has magnitude ∫₄⁶(x − 4)dx = 2.
Signed integral 6; total geometric area 10 square units.
Look for these in your work
- Signed integral evaluated
- Crossing located
- Positive areas added separately
Common mistakes
- Subtracting F(b) − F(a) in reverse order
- Treating area below the axis as positive signed integral
- Integrating bottom minus top for an area
- Forgetting to split an interval if the curve order changes
What can you explain now?
How do you evaluate ∫ₐᵇ f(x)dx from an antiderivative F?
Compare with the explanation
F(b) − F(a)
Substitute the upper limit and subtract the lower-limit value.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
How do signed integrals differ from positive geometric area, and when should you split the interval?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.