6

Multiple integrals

For functions of several variables, just as we may consider derivatives with respect

to two or more of them, so may the integral of the function with respect to more

than one variable be formed. The formal definitions of such multiple integrals are

extensions of that for a single variable, discussed in chapter 2. We first discuss

double and triple integrals and illustrate some of their applications. We then

consider changing the variables in multiple integrals and discuss some general

properties of Jacobians.

6.1 Double integrals

For an integral involving two variables – a double integral – we have a function,

f(x, y) say, to be integrated with respect toxandybetween certain limits. These

limits can usually be represented by a closed curveCbounding a regionRin the

xy-plane. Following the discussion of single integrals given in chapter 2, let us

divide the regionRintoNsubregions ∆Rpof area ∆Ap,p=1, 2 ,...,N, and let

(xp,yp) be any point in subregion ∆Rp. Now consider the sum

S=

∑N

p=1

f(xp,yp)∆Ap,

and letN→∞as each of the areas ∆Ap→0. If the sumStends to a unique

limit,I, then this is called thedouble integral off(x, y)over the regionRand is

written

I=

∫

R

f(x, y)dA, (6.1)

wheredAstands for the element of area in thexy-plane. By choosing the

subregions to be small rectangles each of area ∆A=∆x∆y, and letting both ∆x