Advanced Calculus fi..

Integral Calculusof Functionsof Several VariablesWe review here the properties of the integral for functions of one variable.Let f (x) be defined for a 5 x 5 b. The definite integralis defined as a limit:Here one is considering subdivisions of the interval a 5 x 5 b by values a = xo 0, one can choose 6 > 0 so small that for everysuch subdivision of mesh h less than 6 and no matter how the x: are chosen in theintervals xi-, 5 x 5 xi, one hasThe integral we have defined is called the Riemann integral. It can be shown toexist if, for example, f (x) is continuous for a 5 x 5 b; see Section 4.10. Further,it exists if f is continuous except at a finite number of points, provided that f isbounded: If (x)l 5 M for some constant M, for a 5 x 5 b.21 5