Area under a curve is one of the most predictable question types in O-Level A Math (4049). It shows up almost every year in Paper 2, usually worth 5 to 8 marks, and the marking is generous if you set up the integral correctly. The students who lose marks rarely lose them on the integration itself. They lose them on limits, on signs, and on not knowing which variable to integrate with respect to.
Here is the method we drill with our students at Dojo, written the way we actually solved these papers ourselves.
Step 1: Always sketch first
Do not skip this, even if the question does not ask for a sketch. A rough sketch takes forty seconds and tells you:
- Where the curve cuts the x-axis (these are often your limits)
- Whether any part of the region lies below the x-axis
- Whether the region is bounded by a line, another curve, or the y-axis
- Whether you need to split the region into two parts
Mark the region you are being asked for by shading it lightly. Examiners give method marks based on the integral you write down, so a correct sketch usually leads straight to a correct set-up.
Step 2: Find your limits properly
The two most common sources of limits are:
- Given in the question. For example, "the region bounded by the curve, the x-axis, and the lines x = 1 and x = 4".
- Intersection points. Solve the curve equal to zero for x-axis intercepts, or set the curve equal to the line and solve the resulting equation.
When you solve for intersections, write the working out clearly. If y = x² and y = 3x + 4 intersect, then x² = 3x + 4, so x² − 3x − 4 = 0, giving x = 4 and x = −1. Those two values become your limits. Many students find the x-values and then forget which one is the lower limit. The lower limit is always the smaller x-value when integrating with respect to x.
Step 3: Watch for regions below the x-axis
This is the single biggest mark-loser. A definite integral gives you a signed area. If part of the curve dips below the x-axis, that portion of the integral comes out negative, and adding it to the positive portion gives you the wrong total.
The fix: split the integral at the x-intercept.
If a curve crosses the x-axis at x = 2 and you want the area from x = 0 to x = 5, compute the integral from 0 to 2 and the integral from 2 to 5 separately, then add the modulus of each.
Write it out as: Area = |∫ from 0 to 2| + |∫ from 2 to 5|.
If your final area comes out negative, you have almost certainly forgotten this. An area can never be negative, so a negative answer is a red flag that should make you go back and check.
Step 4: Know when to integrate with respect to y
If the region is bounded by the y-axis rather than the x-axis, you integrate x with respect to y.
The procedure:
- Rearrange the equation to make x the subject
- Convert your limits into y-values by substituting the x-values into the original equation
- Evaluate ∫ x dy between those y-limits
For example, for y = x² + 1, you rearrange to x = √(y − 1) and integrate from y = 1 upwards. Papers often signal this with phrasing like "bounded by the curve, the y-axis and the line y = 5". Read that phrase carefully. It is the examiner telling you which variable to use.
Step 5: Areas between two curves
When a region sits between a curve and a straight line, or between two curves, use:
Area = ∫ (upper function − lower function) dx
between the intersection points. Subtracting in this order means you never have to worry about negative signs, even if the whole region sits below the x-axis.
A useful shortcut: if the region is bounded above by a straight line, you can sometimes find the area of a trapezium or triangle using coordinate geometry and subtract the area under the curve. This is faster and less error-prone when the curve is awkward to integrate. Both methods earn full marks.
Common mark-losing mistakes
- Forgetting to substitute the lower limit, or subtracting in the wrong order
- Losing the constant when integrating terms like (2x + 3)⁵, which needs division by the coefficient of x
- Leaving the answer as a decimal when the question says "exact value", which means keep fractions, surds, e or ln
- Not including units when the question is set in a real-world context
- Writing the integral sign without dx
Practise the set-up, not just the integration
A productive drill: take ten past-paper area questions and, without evaluating anything, just write down the correct definite integral for each. Ten minutes, ten set-ups. Once the set-up is automatic, the integration is routine work you have already mastered from earlier in the syllabus.
If integration still feels shaky, come and work through a few of these with us live. You can start a free trial class with a Dojo tutor on WhatsApp here: https://wa.link/dsgbkf