Integration can be studied on many levels. In Chap. 6, the theory was developed for reasonably well-behaved functions on subintervals of the real line. In Chap. 11 we shall encounter a very highly developed theory of integration that can be applied to much larger classes of functions, whose domains are more or less arbitrary sets, not necessarily subsets of $R^n$. The present chapter is devoted to those aspects of integration theory that are closely related to the geometry of euclidean spaces, such as the change of variables formula, line integrals, and the machinery of differential forms that is used in the statement and proof of the n-dimensional analogue of the fundamental theorem of calculus, namely Stokes’ theorem.

10.1 Definition

Suppose $I^k$ is a k-cell in $R^k$, consisting of all
$x = \left(x_1,\cdots ,x_n\right)$ such that

$$(1)\text{ }a_i\leq x_i\leq b_i\text{ }(i = 1,\cdots ,k) ,$$

$I^j$ is the j-cell in $R^j$ defined by the first j inequalities (1), and f is a real continuous function on $I^k$.
    Put $f=f_k$, and define $f_{k-1}$ on $I^{k-1}$ by

$$f_{k-1}\left(x_1, \cdots , x_{k-1}\right)=\int _{a_k}^{b_k}f_k\left(x_1,\cdots ,x_{k-1},x_k\right)\text{ dx}_k.$$

The uniform continuity of $f_k$ on $I^k$ shows that $f_{k-1}$ is continuous on $I^{k-1}$l.
Hence we can repeat this process and obtain functions $f_j$, continuous on $I^j$, such that $f_{j-1}$ is the integral of $f_j$, with respect to $x_j$, over $\left[a_j, b_j\right]$. After k steps we
arrive at a number $f_0$, which we call the integral of f over $I^k$; we write it in the form

$$(2)\text{ }f_{I^k}f(x)\text{ dx} \text{ or} f_{I^k}f.$$

A priori, this definition of the integral depends on the order in which the k integrations are carried out. However, this dependence is only apparent. To prove this, let us introduce the temporary notation $L(f)$ for the integral (2) and$L'(f)$ for the result obtained by carrying out the k integrations in some
other order.