7.6 Properties of Definite Integrals

Notes

For a function $f$ that can be integrated on the intervals involved:

• $\displaystyle \int_a^a f\,dx = 0$ and $\displaystyle \int_b^a f\,dx = -\int_a^b f\,dx$.
• $\displaystyle \int_a^b c\,f\,dx = c\int_a^b f\,dx$ and $\displaystyle \int_a^b (f + g)\,dx = \int_a^b f\,dx + \int_a^b g\,dx$.
• Additivity: $\displaystyle \int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx$.
• Example: if $\displaystyle \int_1^4 f = 7$ and $\displaystyle \int_4^6 f = 3$, then $\displaystyle \int_1^6 f = 10$.