Calculus/Multivariable Calculus/Riemann Sums and Iterated Integrals

Riemann sums

When looking at these forms of integrals, we look at the Riemann sum. Recall in the one-variable case we divide the interval we are integrating over into rectangles and summing the areas of these rectangles as their widths get smaller and smaller. For the multiple-variable case, we need to do something similar, but the problem arises how to split up R2, or R3, for instance.

To do this, we extend the concept of the interval, and consider what we call a n-interval. An n-interval is a set of points in some rectangular region with sides of some fixed width in each dimension, that is, a set in the form {xRn|aixibi with i = 0,...,n}, and its area/size/volume (which we simply call its measure to avoid confusion) is the product of the lengths of all its sides.

So, an n-interval in R2 could be some rectangular partition of the plane, such as {(x,y) | x ∈ [0,1] and y ∈ [0, 2]|}. Its measure is 2.

If we are to consider the Riemann sum now in terms of sub-n-intervals of a region Ω, it is

${\displaystyle \sum _{i;S_{i}\subset \Omega }f(x_{i}^{*})m(S_{i})}$

where m(Si) is the measure of the division of Ω into k sub-n-intervals Si, and x*i is a point in Si. The index is important - we only perform the sum where Si falls completely within Ω - any Si that is not completely contained in Ω we ignore.

As we take the limit as k goes to infinity, that is, we divide up Ω into finer and finer sub-n-intervals, and this sum is the same no matter how we divide up Ω, we get the integral of f over Ω which we write

${\displaystyle \int _{\Omega }f}$

For two dimensions, we may write

${\displaystyle \int \int _{\Omega }f}$

and likewise for n dimensions.

Iterated integrals

Thankfully, we need not always work with Riemann sums every time we want to calculate an integral in more than one variable. There are some results that make life a bit easier for us.

For R2, if we have some region bounded between two functions of the other variable (so two functions in the form f(x) = y, or f(y) = x), between a constant boundary (so, between x = a and x =b or y = a and y = b), we have

${\displaystyle \int _{a}^{b}\,\int _{f(x)}^{g(x)}h(x,y)\,dydx}$

An important theorem (called Fubini's theorem) assures us that this integral is the same as

${\displaystyle \int \int _{\Omega }f}$ ,

if f is continuous on the domain of integration.