02 Properties of the definite integral
1Zero width
Statement
- ∫aa f(x) dx = 0
Use
- Limits equal → the integral vanishes
2Reversal
Statement
- ∫ab f = −∫ba f
Use
- Swapping limits changes the sign
3Additivity
Statement
- ∫ab f + ∫bc f = ∫ac f
Use
- Split at a point where the rule changes (e.g.
- |x|)
4Constant multiple
Statement
- ∫ k·f = k∫ f
Use
- Take constants outside
5Linearity
Statement
- ∫ (f ± g) = ∫ f ± ∫ g
Use
- Integrate term by term
6Even and odd
Statement
- ∫−aa f = 0 if f is odd; 2∫0a f if f is even
Use
- Saves a full evaluation
Worked example with an absolute value
- ∫−12 |x| dx = ∫−10 (−x) dx + ∫02 x dx = ½ + 2 = 5/2.
- Splitting at x = 0 is compulsory; integrating |x| as if it were x gives the wrong value.