Area between two curves is one of the most predictable question types in Paper 2 of O-Level Additional Mathematics (4049). It looks intimidating because the diagram is often shaded and messy, but the method is almost always the same four steps. Once you drill it, these questions become some of the fastest marks on the paper.
The Core Idea
If one graph sits above another between two x-values, the area of the region trapped between them is:
Area = ∫ from a to b of (top curve minus bottom curve) dx
That is it. The only real skills being tested are:
- Finding the limits a and b, usually the points of intersection.
- Deciding which function is on top.
- Integrating correctly.
- Substituting limits without sign errors.
Step by Step With a Worked Example
Find the area of the region enclosed by the curve y = x² and the line y = x + 2.
Step 1: Find the Points of Intersection
Set them equal:
x² = x + 2 x² − x − 2 = 0 (x − 2)(x + 1) = 0 x = −1 or x = 2
These are your limits. Never guess limits from the diagram. Examiners award a method mark for correctly solving the simultaneous equations.
Step 2: Decide Which Is on Top
Pick any x value between −1 and 2, say x = 0. The line gives y = 2, the curve gives y = 0. The line is above the curve, so the integrand is (line − curve).
Step 3: Set Up and Integrate
Area = ∫ from −1 to 2 of (x + 2 − x²) dx = [x²/2 + 2x − x³/3] from −1 to 2
Step 4: Substitute Carefully
At x = 2: 2 + 4 − 8/3 = 10/3 At x = −1: 1/2 − 2 + 1/3 = −7/6
Area = 10/3 − (−7/6) = 27/6 = 4.5 units²
Notice how the second bracket is subtracted as a whole. Losing the bracket is the single most common careless error we see in marked scripts.
When the Curves Cross in the Middle
If the two graphs swap positions inside your interval, you cannot use one integral. Consider y = x³ and y = x. They meet at x = −1, 0 and 1. Between −1 and 0 the cubic is above the line, and between 0 and 1 the line is above the cubic.
Split it:
Area = ∫ from −1 to 0 of (x³ − x) dx + ∫ from 0 to 1 of (x − x³) dx = 1/4 + 1/4 = 0.5 units²
If you had integrated straight through from −1 to 1, the two halves would cancel and you would write 0, which earns almost nothing. Always sketch or test a value in each sub-interval.
Integrating With Respect to y
Some regions are far easier with horizontal strips. Take the curve y² = x and the line y = x − 2.
Rewrite both in terms of y: x = y² and x = y + 2. Solve y² = y + 2 to get y = −1 and y = 2.
Area = ∫ from −1 to 2 of (y + 2 − y²) dy = 4.5 units²
Doing this one in terms of x would force you to split the region into two parts because of the two branches of the square root. Look for this whenever the boundary curve is written as y² = something, or when the region is bounded on the left and right rather than top and bottom.
Shortcuts Examiners Accept
When one boundary is a straight line, you can often replace an integral with the area of a triangle or trapezium.
- Area under the line from a to b can be computed with ½ × base × height or ½(sum of parallel sides) × width.
- Subtract or add the area under the curve as required by the shaded diagram.
This is fully acceptable in the mark scheme and is usually faster. Just make sure you state clearly what you are subtracting from what.
Common Mistakes to Avoid
- Writing (curve − line) when the line is above, giving a negative answer. If you get a negative area, swap the order rather than just deleting the minus sign silently.
- Forgetting to find intersections and instead reading approximate limits off the sketch.
- Not splitting when curves cross.
- Dropping the constant when integrating, or integrating (x + 2 − x²) term by term but mis-signing the x³/3.
- Leaving out units². Some questions specify units, so match the phrasing in the question.
- Rounding too early. Keep exact fractions until the final line, then give exact or 3 significant figures as asked.
A Quick Checklist for the Exam
- Sketch or label the diagram, mark the intersections.
- Solve simultaneously for the limits.
- Test one x value to confirm which graph is on top.
- Write the full integral with limits before integrating. This secures the method mark even if the arithmetic slips.
- Substitute with brackets, simplify to an exact fraction, and state units².
Practise five of these back to back from past papers and you will start recognising the setup within seconds of reading the question.
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