5.3 Properties of definite integrals

For definite integrals we can, using the fundamental theorem of calculus, determine quite a few properties.

1. {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{a}f(x)\kern 1.66702pt dx = 0.
2. {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{b}^{a}f(x)\kern 1.66702pt dx = −{\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{b}f(x)\kern 1.66702pt dx. The definite integral is not just an area!
3. {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{b}f(x)\kern 1.66702pt dx ={\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{c}f(x)\kern 1.66702pt dx +{\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits }_{c}^{b}f(x)\kern 1.66702pt dx. The value of c is arbitrary, it doesn’t have to be between a and b!
4. {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{b}f(x)dx ≥ 0\text{ if }f(x) ≥ 0.
5. If m ≤ f(x) ≤ M and b ≥ a, then m(b − a) ≤{\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{b}f(x)dx ≤ M(b − a).