Integration
What this domain covers
Integration in 8227 covers reversing differentiation, evaluating definite integrals, finding areas under curves and between curves, and volumes of revolution about the x-axis.
Core rules
- ∫x^n dx = x^(n+1)/(n+1) + c for n ≠ −1; ∫x^(−1) dx = ln|x| + c; ∫sin x dx = −cos x + c; ∫cos x dx = sin x + c; ∫e^x dx = e^x + c.
- Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a); the constant cancels, so no + c is needed.
- Area: if f(x) < 0 on part of the interval, split at the root and add the magnitudes, because a single integral cancels positive and negative regions.
- Area between curves: integrate (upper − lower) between the intersection x-values.
- Volume about the x-axis: V = π ∫ from a to b of y² dx.
Worked example
Find the area enclosed by y = x and y = x². The curves meet where x = x², that is at x = 0 and x = 1. On 0 < x < 1 the line is above the parabola, so the area is ∫ from 0 to 1 of (x − x²) dx = [x²/2 − x³/3] from 0 to 1 = ½ − ⅓ = ⅙.
Traps
- Omitting + c in an indefinite integral.
- Forgetting to square y, or forgetting π, in a volume of revolution.
- Not splitting the interval at a root, so two lobes cancel to zero.
- Substituting the limits the wrong way: F(a) − F(b) instead of F(b) − F(a).
- Integrating y rather than y² when the question asks for a volume.