Integration and Applications
Core ideas
Candidates must integrate powers of x, obtaining x^(n+1)/(n+1) + c for n ≠ −1, and integrate polynomials term by term. Definite integrals evaluate area under a curve between limits. Areas between a curve and a line, or between two curves, require subtracting the lower function from the upper. Also examinable: the constant of integration, the trapezium rule for approximate area, and using a given gradient function to recover the original curve by integration with a boundary condition.
Traps
- Omitting + c in indefinite integrals.
- Forgetting that areas below the x-axis give negative integrals; take the modulus or split the region.
- Subtracting limits in the wrong order.
- Losing accuracy in the trapezium rule by using too few strips.
- Not applying the boundary condition to find c after integrating.
Worked example
Find ∫₁³ (2x + 1) dx. The antiderivative is x² + x. Evaluating: (9 + 3) − (1 + 1) = 12 − 2 = 10. So the area is 10 square units.
Rule of thumb
For any area question, sketch first: identify which curve is on top, then integrate (upper − lower) between the intersection limits.