Integration questions in Paper 1 reward careful setting up. The reports show that students usually know that they need to integrate. The marks go missing in the limits, the constant, the reverse chain rule, and the way squares are expanded for volumes. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Forgetting the constant of integration. This is described as common when finding a curve from its gradient, or when integrating twice. Without the constant, the later marks cannot be earned.
- Wrong limits. Using the answer to part (a) as an upper limit, using the coordinates of another point or the y-intercept as limits, setting the lower limit to 0 when the diagram shows otherwise, and using limits for x with a function of y.
- Not showing the limits being substituted. If the integral is incorrect and the substitution of both limits is not shown, no method marks can be given. Evaluating the integral on a calculator earns no credit for method.
- Sign and subtraction slips. Subtracting the lower from the upper limit incorrectly, or evaluating (−1)³ as −1. A negative area given as the final answer does not score the final mark.
- The reverse chain rule. For an integral of the form (ax + b)ⁿ, divide by the coefficient of x as well as by the new power. Reports mention students who divided by the power only, or who forgot to divide the first term by 4.
- Squaring for volumes. To find a volume of revolution, you integrate πy². Squaring a function with two terms gives three terms. Students obtained only two terms, or squared the whole function incorrectly. Others found the area under the curve instead of the volume.
- Area between curves and volume between curves. For the volume between two curves, the reports note that only some students subtracted the integrals of the squares of the two functions. A few found the volume of the region formed by a straight line using the formula for a frustum of a cone, which also worked.
- Rewriting the integrand badly. Attempts to form a common denominator and then integrate a result that could not be integrated, or writing the integral of a sum of squares as the integral of a square of a sum.
What examiners advise
- Show the substitution of both limits and the subtraction.
- Always include the constant and find it using a given point.
- Read the question carefully to see whether it wants an area or a volume, and which axis it is about.
- Draw a diagram when it is not given. Reports link wrong answers with an incorrect or missing diagram.
- Think about the necessary working. Answers supported by a calculator's integral function, with no working shown, gain nothing.
- Check the sign of an area. A negative area given as the final answer did not earn the last mark.
Habits that help
- Differentiate your answer to check it.
- Write the integral with all the terms in power form before integrating.
- Put brackets around negative limits when substituting.
Questions students ask
Why do I lose marks when I use my calculator's integral function?
A calculator answer with no working earns nothing for method. If the integral is incorrect and the substitution of both limits is not shown, no method marks can be given.
Do I always need the constant of integration?
Yes, when you find a curve from its gradient or integrate twice. Without it the later marks cannot be earned. Find it using a given point.
How do I integrate (ax + b)ⁿ?
Use the reverse chain rule: divide by the new power and by the coefficient of x. Reports mention students who divided by the power only.
How do I find the volume of revolution?
Integrate πy², and remember that squaring a function with two terms gives three terms. Some students found the area under the curve instead of the volume. For a volume between two curves, subtract the integrals of the squares.
My area came out negative. What should I do?
Check the limits and the subtraction. A negative area given as the final answer does not earn the final mark.
Which limits do I use?
Use limits that match the variable and the diagram. Reports record the answer to part (a) used as an upper limit, the y-intercept used as a limit, and limits for x used with a function of y. Draw a diagram if one is not given.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.