|
The Riemann Double Integral
In this chapter we return to the discussion of the Riemann Integral. We begin by proving some further properties of the integral and then proceed to discuss a rigorous definition of the double integral. This will involve a treatment of the conditions that a curve must satisfy in order that it may possess a length.
Change of Variable
We shall now justify the method of change of variable. We begin by proving the following lemma:
If f(x), g(x) are integrable over (a,b) so is f(x)g(x)
Let m1, M1 ; m2, M2 ; m, M be the bounds of f,g,fg respectively in any sub-interval of (a,b) and let H, K (as shown on the next page) be the upper bounds of |f|,
|g| in (a,b) given any E we can find X Y points of the sub-interval such that -
|
|