3-4

3.011 Suppose that f(x) is integrable^{†} over the largest of the intervals (p, q), (p, r), (r, q). Then (depending on the relative positions of the points p, q, and r) it is also integrable over the other two intervals and we have

${\int}_{p}^{q}f\left(x\right)\text{d}x={\int}_{p}^{r}f\left(x\right)\text{d}x+{\int}_{r}^{q}f\left(x\right)\text{d}x$

FI II 126

3.012The first mean-value theorem. Suppose (1) that f(x) is continuous and that g(x) is integrable over the interval (p, q), (2) that m ≤ f(x) ≤M and (3) that g(x) does not change sign anywhere in the interval (p, q). Then, there exists at least one point ξ (with p ≤ ξ ≤ q) such ...

Start Free Trial

No credit card required