Integration and exact area
An antiderivative reverses differentiation; a definite integral measures signed accumulation.
For n ≠ −1, an antiderivative of xⁿ is xⁿ⁺¹/(n + 1). The exceptional power x⁻¹ integrates to ln|x| on intervals not crossing zero. Add a constant to an indefinite integral.
A definite integral uses F(b) − F(a). Area beneath a curve requires attention to sign: split at crossings of the axis and add positive areas. Differentiating an antiderivative checks it; numerical integration alone cannot establish an exact result.
Worked example
Evaluate the integral of 3x² from 1 to 2.
- An antiderivative is x³.
- Evaluate 2³ − 1³.
Answer: 7
Revise first: Differentiating powers.
Practise integration and exact area
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 9FM0 · core · 5 Further calculus: Integration methods
- 7367 · core · E Further calculus: Integration methods
- H245 · core · 6 Further calculus: Integration methods
- H645 · core · Further calculus: Integration methods
- 9FM0 · core · 5 Further calculus: Exact polynomial definite integrals
- 7367 · core · E Further calculus: Exact polynomial definite integrals
- H245 · core · 6 Further calculus: Exact polynomial definite integrals
- H645 · core · Further calculus: Exact polynomial definite integrals
- 6993 · core · CA8–CA11: Exact polynomial definite integrals
- 4PM1 · core · 9A–9C: Exact polynomial definite integrals
- 9FM0 · core · 5 Further calculus: An exact integral of an even sine power
- 7367 · core · E Further calculus: An exact integral of an even sine power
- H245 · core · 6 Further calculus: An exact integral of an even sine power
- H645 · core · Further calculus: An exact integral of an even sine power
Next practice: Areas in polar coordinates.