Properties of the definite integral.

Proposition 8.4.1   $ \int_a^a f(x) \; dx =0$ .

Proposition 8.4.2   Let $ f$ be a function integrable on an interval $ I$ and let $ a,b,c \in I$ . Then:

$\displaystyle \int_a^b f(x) \; dx = \int_a^c f(x) \; dx + \int_c^b f(x) \; dx.$    

Proposition 8.4.3   Suppose that the functions $ f$ and $ g$ are integrable on $ [a,b]$ . Then $ f+g$ is integrable on $ [a,b]$ and:

$\displaystyle \int_a^b [f(x)+g(x)] \; dx = \int_a^b f(x) \; dx + \int_a^b g(x) \; dx.$    

Proposition 8.4.4   Suppose that the function $ f$ is integrable on $ [a,b]$ . Then $ \alpha f$ is integrable on $ [a,b]$ and:

$\displaystyle \int_a^b \alpha f(x) \; dx = \alpha \int_a^b f(x) \; dx.$    

Proposition 8.4.5   If $ f$ is integrable on $ [a,b]$ and if $ f(x) \geq 0$ on $ [a,b]$ , then $ \int_a^b f(x) \; dx \geq 0$ .

Corollary 8.4.6   If $ f$ and $ g$ are integrable on $ [a,b]$ and if $ f(x) \geq g(x)$ on $ [a,b]$ , then $ \int_a^b f(x) \; dx \geq \int_a^b g(x) \; dx$ .

Corollary 8.4.7   If $ f$ is integrable on $ [a,b]$ , then $ \vert f\vert$ is also integrable on $ [a,b]$ and $ \left\vert \int_a^b f(x) \; dx \right\vert \leq \int_a^b \vert f(x)\vert \; dx$ .

For definite integrals, the method of substitution (v.s. 2.2) is described as follows:

Theorem 8.4.8   Let $ a,b$ be two numbers such that $ a<b$ Let $ g$ be a function having a continuous first derivative on $ [a,b]$ and such that $ a \leq x \leq b \Longrightarrow
m \leq g(t) \leq M$ . If $ f$ is a function, continuous on the interval $ [m,M]$ , then for any $ u \in [a,b]$ we have:

$\displaystyle \int_{g(a)}^{g(u)} f(t) \; dt = \int_a^u f[g(x)] \cdot g'(x) \; dx$    

In particular

$\displaystyle \int_{g(a)}^{g(b)} f(t) \; dt = \int_a^b f[g(x)] \cdot g'(x) \; dx$    

Noah Dana-Picard 2007-12-28