11
Line and multiple integrals
In Chapter 7 we extended the discussion of differentiation given in Chapter 3 to functions of several variables. In this chapter we will extend the discussion of integration given in Chapter 4 in a similar way. We will begin by discussing functions of two variables, which we will usually take to be the Cartesian co-ordinates x, y, although they could equally well be, for example, a position and a time. The discussion will then be generalised to three or more variables and to other co-ordinate systems, especially polar co-ordinates in three dimensions. This will form the basis for important applications in vector analysis, which is an essential tool in understanding topics such as electromagnetic fields, fluid dynamics and potential theory, and which will be discussed extensively in Chapter 12.
11.1 Line integrals
In this section, we first introduce line integrals and their properties in two dimensions and then briefly indicate their extension to three dimensions, which is relatively straightforward. In both cases we will use Cartesian co-ordinates.
11.1.1 Line integrals in a plane
Suppose y = f(x) is a real single-valued monotonic continuous function of x defined in some interval x1 < x < x2, as represented by the curve C shown in Figure 11.1a. Then, if P(x, y) is a real single-valued continuous function of x and y for all points on the curve C, the integral
is called a line integral and the symbol C on the integration sign indicates that the path, or contour, of integration from the initial point A to the end point B is along the curve C. The formal definition of a line integral is closely related to that of ordinary integrals as discussed in Chapter 4. Thus for a function P(x, y),
(11.2) 
where the sum is over all elements δxi on the curve C. Since y = f(x), the integral (11.1) is equivalent to an ordinary integral with respect to a single variable x. Thus,
Figure 11.1 Integration path for (a) a single-valued function and (b) a two-valued function.
We could also consider the integral along C as being with respect to the variable y by inverting the relation y = f(x) to give x as a function of y along C, for example x = g(y). Then if Q(x, y) is another real single-valued continuous function of x and y for all points on the curve C, a line integral analogous to (11.3) is
Alternatively, we can convert line integrals over x and y into line integrals over y and x by writing
(11.5) 
and
(11.6) 
where we have used dx = g′(y)dy and dy = f′(x)dx respectively. In what follows, it is often useful to write line integrals along a given curve C in the general form
where P and Q are given functions.
In the above discussion, we have assumed that the contour of integration C can be described by a single-valued function y = f(x). This is not always the case. Consider the curve shown in Figure 11.1b. For some values of x, two different values of y are obtained, that is, f(x) is not single-valued. In this case, the integral must be divided into two parts (or more if f(x) is multi-valued) in each of which it is single-valued, and by the results of Chapter 4, we may write
where C1 is the path from A to D, which is described by the function f1(x), and C2 is the path from D to B, which is described by the function f2(x).
Finally, the path C may be defined by an implicit relationship between the x and y co-ordinates, and in particular by parametric forms x = x(t) and y = y(t). Here, both x and y are defined by single-valued differentiable functions of a single parameter t, so that as t goes from tA, the value of t at A, to tB, the value of t at B, the path between A and B is traced out in the right direction once and once only. Any line integral of the general form (11.7) can then be transformed into a definite integral over t:
(11.8) 
by substituting the given forms x(t), y(t).
11.1.2 Integrals around closed contours and along arcs
In Figure 11.1a the integral is along the path C from A to B, and it is clear that the value of the line integral depends on the functional form y = f(x) of the curve, and so in general the integral will be different for different paths between the same two points, although later we shall meet examples where this is not true and the line integral only depends on the end points of the integral. We could of course also take the integration from B to A. It follows from the results established in Chapter 4 for ordinary integrals that
(11.9) 
where it is understood that the integration is still along the path C, but in the reverse direction. It then follows that a line integral from A to B, and returning to A along the same path is zero. However, for a closed contour where the return path from B to A is not the same as that from A to B, in general the integral is non-zero, although again we will meet examples later where this is not true.
If the integration path is a simple closed plane curve, that is, one that does not cross itself, such as that shown in Figure 11.3, the integral is written
Figure 11.3 A simple closed plane curve.
It is conventionally assumed that the integration is in the counter-clockwise direction, but to be totally unambiguous, the direction of travel around the closed contour can be indicated by an arrow on the circle, i.e.
where the symbols indicate integration in the counter-clockwise (positive) and clockwise (negative) directions, respectively. A closed curve cannot be represented by a single-valued function, so when evaluating integrals like (11.10), the technique of breaking the contour of integration into sections must be used.
We may also consider line integrals of the form
where dl is an infinitesimal arc length of the curve C. For the simple case P(x, y) = 1, the integral
gives the length of the curve f(x) from A to B. The integrals (11.11a) and (11.11b) may be converted to the standard form (11.1), with a modified function P, by using the result
where y = f(x), or
(11.12b) 
if x and y are given in parametric forms as functions of a parameter t.
11.1.3 Line integrals in three dimensions
The extension of the above ideas to functions of three real variables x, y, z is straightforward, and will be summarised here very briefly. In an obvious notation, the general line integral (11.7) becomes
where P, Q, R are single-valued functions of x, y, z, and if y = f1(x), z = f2(x), with x1 < x < x2 along the path of integration, then
(11.14) 
in analogy with (11.3), with similar expressions analogous to (11.4) for the other terms in (11.13). Alternatively, if the path is specified by three functions x(t), y(t), z(t) of a single parameter t, with tA < t < tB, then (11.13) becomes a single integral
(11.15) 
Finally, we may again consider integrals of the form
(11.16a) 
in analogy to (11.11), where the element of arc length dl is now given by
(11.16b) 
or, if the path C is specified by a real parameter t,
11.2 Double integrals
The ideas discussed in Chapter 4 for defining and evaluating definite integrals may be extended to evaluate integrals over two or more variables. We start by considering double integrals, that is, integrals over two variables, again using Cartesian co-ordinates. These may be written in a number of forms:
where S is an area, which we will assume is defined by a simple boundary curve, that is, one that does not cross itself, and where the limits on the x and y integrations will be specified shortly. In addition, we will assume that the function f(x, y) is finite, single-valued and continuous within and on the boundary.
In Chapter 4 we defined a definite integral of a function f(x) of a single variable x by dividing the range of integration into n small intervals of width δxn, and then taking the limit
where f(xn) is the value of f(x) at the mid-point of the interval. The double integrals (11.17a) are defined in an analogous way by dividing S into small elements by a grid of lines parallel to the x and y axes, as shown in Figure 11.5a. If the grid widths are j1, j2, …, jr in the x direction, and k1, k2, …, ks in the y direction, the area of a rectangle rs is jrks. If f(xr, ys) is a point within this rectangle, the double integral is defined as a sum of contributions
(11.17b) 
in the limit that all j r and ks tend to zero, in which case the number of rectangles tends to infinity. This sum can be conveniently rewritten in the form
where, for fixed s, the sum over r is the contribution from the horizontal shaded strip in Figure 11.5a and ys can be assumed to be constant for all terms in the sum. In the limit jr → 0, this sum is the integral
where the limits of integration are shown in Figure 11.5b. The double sum then becomes
where β1 and β2 are the minimum and maximum values of y in the region S. Thus the double integral is
Figure 11.5 Constructions for defining a double integral.
Alternatively, we could have done the sum over s first, followed by the sum over r. In this case the double integral would be
where α1 and α2 are the minimum and maximum values of x in the region S, as shown in Figure 11.4b.
Interchanging the order can often be useful in simplifying the integrations that have to be performed, and is usually valid. However, one should remember that, in the above discussion, we have assumed that the integrand f(x, y) is continuous and finite within and on the boundary of the region of integration S. If this condition is not satisfied the integrals (11.18a) and (11.18b) may or may not exist; and if they both exist they may or may not be equal. An example of the latter behaviour is the integral,
where the region of integration S is bounded by the lines x = 0, x = 1, y = 0, y = 1. The integrand has a discontinuity on the boundary of S at the point (0, 0) and thus violates the above condition, so that it is not necessarily safe to invert the two integrations. This is confirmed by setting y = u − x, when it is easily shown that I = 1/2. However, inverting the order of integration gives
and using the same substitution gives I′ = −1/2.
11.2.1 Green's theorem in the plane and perfect differentials
It is quite common for a line integral to be taken around a closed loop and we have seen in Section 11.1.1 how to evaluate such integrals. Green's theorem in the plane shows how to relate them to double integrals over the region enclosed by the loop, which is often easier to evaluate.
Let P(x, y) and Q(x, y) be two functions of x and y with continuous, finite partial derivatives in a region R and on the boundary C, as shown in Figure 11.8. Then
where y1(x) is the curve STU and y2(x) is the curve SVU. Evaluating the right-hand side gives
where the notation in the final integral means the integral is around the closed curve C. In an analogous way, if we start with the integral
and let x1(y) be the curve TSV and x2(y) be the curve TUV, we have
Figure 11.8 Figure used in the derivation of Green's theorem in the plane.
Subtracting (11.19) from this equation gives
(11.20) 
which is Green's theorem in the plane.
Green's theorem in the plane shows that a line integral of the general form (11.7), where C is a loop, can be converted to a double integral over the area enclosed by the loop. It also shows that if
then the line integral around the loop vanishes, i.e.
(11.22a) 
Equation (11.21a) is also the condition that
is an exact, or perfect, differential (cf. Section 7.2.2) with
(11.21c) 
Hence if (11.21a) is satisfied, the line integral from A → B along any path is given by
(11.22b) 
where IA and IB are the values of I at the points A and B, respectively, independent of the path connecting A to B.
To summarise, the necessary and sufficient condition for any loop integral (11.8c) to vanish for a closed loop and for the integral (11.22d) to be independent of the path for all paths is that (11.21b) is a perfect differential. This result extends to three dimensions, that is, the general line integral in three dimensions (11.13) is also independent of the path if
is a perfect differential, that is, if [cf. (7.19b)]
is satisfied.
11.2.2 Other co-ordinate systems and change of variables
Up to now we have used mainly the Cartesian system of co-ordinates, but in real applications it is often useful to take advantage of any symmetry the system may have by choosing a different co-ordinate system. Consider the example shown in Figure 11.9a. The shaded area corresponds to the ranges x0 ≤ x ≤ x1, y0 ≤ y ≤ y1; and in Figure 11.9b the shaded area corresponds to either |x| ≤ a, |y| ≤ (a2 − x2)1/2, or |y| ≤ a, |x| ≤ (a2 − y2)1/2. The latter illustrates that in general the ranges of the two variables are not independent. However, had we used plane polar co-ordinates (r, θ), then the shaded area would correspond to the ranges 0 ≤ r ≤ a, 0 ≤ θ < 2π, which are independent. This illustrates the usefulness of choosing co-ordinates to fit the specific problem, and we will see that the evaluation of double integrals like (11.7) can sometimes be considerably simplified if appropriate co-ordinates can be found. However, in order to do this, it is necessary to show how such double integrals can be expressed in variables other than Cartesian co-ordinates.
Figure 11.9 Two co-ordinate systems.
To do this, let us suppose we are using co-ordinates u1, u2 such that the corresponding Cartesian co-ordinates are given by continuous, differentiable functions x(u1, u2) and y(u1, u2). Such variables are called curvilinear co-ordinates because fixing u1 and allowing u2 to vary leads to a family of curves in the x–y plane, as shown in Figure 11.10, and fixing u2 while u1 varies leads to a different family of curves, also shown in Figure 11.10. The value of a function f(x, y) at any point can be expressed in terms of curvilinear co-ordinates, i.e.
and a double integral of f(x, y) over the area S bounded by the curve in Figure 11.10 is given by
where the δSrs are the small areas bounded by ui and ui + δuiwhere i = 1, 2 as shown in Figure 11.10.
Figure 11.10 Curvilinear co-ordinates in a plane, showing lines of constant u1 and u2, spaced by δu1 and δu2, respectively. The area S to be integrated over is the interior of the closed loop and the shaded region is one of the areas δSrs.
In the limit where the separations δu1 and δu2 between such curves tend to zero, the shaded area shown in Figure 11.10 becomes a parallelogram, and to evaluate (11.24) we need to find its area. Referring to Figure 11.11, we write
Figure 11.11 Construction to define the area δSrs in the limit that δu1 and δu2 become infinitesimally small.
If δx1 is the displacement in the x direction, then
and similarly for δy1. So
and the area of the parallelogram δSrs is then given by |δr 1||δr 2|sin θ, where θ is the angle between δr1 and δr2. Hence
(11.25a) 
where the determinant
(11.25b) 
is called the Jacobian and is also written in the shorthand form
(11.25c) 
The sum (11.24) now becomes
and we finally obtain
where the ranges of u1 and u2 are chosen to span S, and |J | is the two-dimensional analogue of the factor dx/du that occurs in a one-dimensional integral when the variable is changed from x to u.
11.3 Curvilinear co-ordinates in three dimensions
Before extending the discussion to include triple integrals, it will be convenient to consider co-ordinate systems other than Cartesian co-ordinates in three dimensions. To do this, we suppose that we have three co-ordinates u1, u2, u3, such that the Cartesian co-ordinates are given by single-valued differentiable functions x(u1, u2, u3), y(u1, u2, u3), and z(u1, u2, u3), and each set of values u1, u2, u3 corresponds to a single point in space:
(11.28) 
where i, j, k are as usual unit vectors along the x, y, z axes, respectively. Alternatively, we can define unit vectors
(11.29) 
so that if
we have
where
The unit vectors ei in general depend on the position r, as we shall shortly demonstrate by example, and since they act as basis vectors at each r, they are written without ‘hats’, even though they are unit vectors. Finally, if
at all r, then u1, u2, u3 are called orthogonal curvilinear co-ordinates and it follows from (11.30) and (11.31) that
(11.32a) 
Similarly, the parallelepiped with adjacent sides given by
reduces to a cuboid with volume
(11.32b) 
if the co-ordinates are orthogonal. This is called the element of volume and plays a crucial role in evaluating triple integrals in orthogonal curvilinear co-ordinates, as we shall see in Section 11.4.1.
We shall now illustrate these ideas by introducing the two most important examples of orthogonal curvilinear co-ordinates: cylindrical and spherical polar co-ordinates, which are used for situations with cylindrical or spherical symmetry, respectively.
11.3.1 Cylindrical and spherical polar co-ordinates
Cylindrical polar co-ordinates in three dimensions are denoted by ρ, φ and z and are shown in Figure 11.12a. They are related to Cartesian co-ordinates by
(11.33a) 
and lie in the ranges
(11.33b) 
Figure 11.12 (a) Cylindrical polar co-ordinates ρ, φ, z, and the associated unit vectors eρ, eφ, ez. The vector eρ is in the direction of the radius vector ρ; eφ is in the xy–plane, tangential to the circle through P, and in the direction of increasing φ; ez is in the z-direction. The three vectors eρ, eφ, ez are mutually orthogonal. (b) Spherical polar co-ordinates r, θ, φ, and the associated unit vectors er, eθ, eφ. The vector er is in the direction of the radius vector r; eφ is in the xy–plane, tangential to the circle through P, and in the direction of increasing φ; eθ is at right angles to er in the direction of increasing θ. The three vectors er, eθ, eφ are mutually orthogonal.
The position vector
(11.34) 
and identifying u1, u2, u3 with ρ, φ, z, one finds, in an obvious notation,
while (11.30a) and (11.30b) give
(11.36) 
Note the factor ρ in the second term. Thus, unlike the case of Cartesian co-ordinates, if φ → φ + dφ for fixed ρ and z, the distance moved is not dφ, but ρ dφ. Another difference from Cartesian co-ordinates is that the basis vectors (11.35), which are also shown in Figure 11.12a, are not constants, but depend on the position r. However, one easily verifies using (11.35) that they are orthogonal,
(11.37) 
Finally, because the basis vectors are orthogonal, the parallelepiped defined by the vectors
is actually a cuboid, as shown in Figure 11.13a, with a volume given by
Figure 11.13 The volume element in (a) cylindrical polar co-ordinates; (b) spherical polar co-ordinates.
This is called the volume element in cylindrical polar co-ordinates.
Spherical polar co-ordinates in three dimensions are (r, θ, φ) and are shown in Figure 11.12b; r = |r| is called the radial co-ordinate, θ is the polar angle between r and the z-axis; and φ is the azimuthal angle. As can be seen, they are related to Cartesian co-ordinates by
and are restricted to the ranges
(11.39b) 
in order to cover the space once, except for the origin, which is given by (r, θ, φ) = (0, θ, φ) for any θ and φ. The position vector is now
(11.40) 
so that using (11.27), one finds, in an obvious notation,
(11.41) 
while
(11.42) 
The unit vectors are shown in Figure 11.12b. They are again orthogonal, so that
(11.43) 
Similarly, the volume element 
is the volume of the cuboid defined by the vectors
and is given by
It is shown in Figure 11.13b.
11.4 Triple or volume integrals
We turn next to triple or volume integrals, denoted by
(11.45) 
where Ω is the region of space to be integrated over and f(x, y, z) is continuous, single-valued and finite within and on the boundary of the region. Since they are a direct generalisation of double integrals, we shall discuss their properties rather briefly.
In Section 11.2.1, we defined double integrals by dividing the region of integration S into small rectangles of side lengths jr, ks, as shown in Figure 11.5, and taking the limit of the weighted sum (11.17a) as both jr and ks tend to zero. Triple integrals are defined in a similar way by dividing Ω into small cuboids with sides of lengths jr, ks and lt, and taking the limit of
as jr, ks and lt tend to zero, where xr, ys, zt is any point within the cuboid r, s, t. As in the two-dimensional case, the order of summation determines the order of integration in the final expression. In particular, if we sum over t, then s, then r, we obtain
(11.46) 
Here α2 and α1 are the maximum and minimum values of x in the region Ω, y2(x) and y1(x) are the maximum and minimum values of y at fixed x in the region Ω, and z2(x, y) and z1(x, y) are the maximum and minimum values of z at fixed values of x and y in the same region. Other orderings of the summation lead to different orderings of the x, y and z integrations, with appropriate limits, but provided f(x, y, z) is single-valued, finite and continuous, they all yield the same value for the integral. Finally, it follows directly from this definition that
(11.47) 
is the volume of the region Ω.
11.4.1 Change of variables
The discussion of changing variables in double integrals given in Section 11.2.2 extends in a straightforward manner to triple integrals, except that instead of summing over infinitesimal parallelograms as in Figures 11.10 and 11.11, we now have to sum over infinitesimal parallelepipeds in three dimensions. We shall not reproduce the derivation but merely state the result, which is a direct generalisation of (11.25) and (11.26). Specifically, if we consider curvilinear co-ordinates u1, u2, u3 (which need not be orthogonal) then
where
(11.48b) 
and the Jacobian
(11.48c) 
Finally, the integrals (11.48a) are often written as
(11.49a) 
without specifying any particular co-ordinate system. However, to evaluate them, a particular co-ordinate system must be chosen with the volume element
which reduces to 
in Cartesian co-ordinates (u1, u2, u3) = (x, y, z). In particular, one easily verifies that (11.49b) is identical to our previous results (11.38) and (11.44) for the volume elements in cylindrical and spherical polar co-ordinates.
Problems 11
11.1 Evaluate the line integral
for two paths: (a) the straight line joining the points A(1, 1) and B(3, 4), and (b) the straight line joining A(1, 1) to C(0, 3), followed by the straight line joining C(0, 3) to B(3, 4).
11.2 Evaluate the line integral
from the point (0, 0) to the point (1, 1) along the curve y = x3: (a) by expressing I(a) as a function of x only and (b) as a function of y only.
11.3 Evaluate the integral
where the contour is the circle x2 + y2 = 1.
11.4 Evaluate the line integral
round the following closed paths, taken to be counter-clockwise: (a) the circle x2 + y2 = 1, (b) the square joining the points (1, 1), ( − 1, 1), ( − 1, − 1) and (1, –1).
11.5 Evaluate the line integral
where the path C is (a) the straight line connecting (0, 0, 0) to (1, 1, 1), and (b) the three connecting straight lines (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1).
11.6 Evaluate the integral
over the triangle bounded by the axes x = 0 and y = 0, and the line x + y = 1.
11.7 Evaluate the integral
by first integrating with respect x and then with respect to y. Then repeat using the reversed order of integration. Comment on your result.
11.8 Invert the order of integration in the double integral
assuming that f(x, y) is well-behaved within the region of integration.
11.9 Reverse the order of integration and hence evaluate the following integrals:
11.10 Evaluate the integral
by reversing the order of integration.
11.11 Evaluate
around the ellipse x2/a2 + y2/b2 = 1.
11.12 Evaluate
around the sides of a square with vertices A(0, 0), B(1, 0), C(1, 1) and D(0, 1) in an anti-clockwise direction. Then convert the line integral to a double integral and verify Green's theorem in a plane.
11.13 Use Green's theorem in the plane to evaluate the integral
from the point (ln 2, 0) to (0, 1) and then to ( − ln 2, 0).
11.14 If the integrands below are perfect differentials, find the values of the integrals between the given points A and B.
11.15 The quantity
is an exact differential. Confirm this by integrating it between the points (0, 0) and (2, 2) along the following paths: (a) y = x2/2, (b) the straight line joining (0, 0) to (2, 0), followed by the straight line joining (2, 0) to (2, 2), (c) the curve defined by the parametric forms x = t2/2 and y = t.
11.16 Integrate the function
over the region of the first quadrant inside the ellipse
using the substitutions x = asin θcos φ, y = bsin θsin φ.
11.17 Evaluate the integral
over the coloured area shown in Figure 11.14, which extends to infinity in the x and y directions.
11.18 Paraboloidal co-ordinates u, w, φ are related to Cartesian co-ordinates by
Find the corresponding unit vectors eu, ew, eφ in terms of i, j, k, and expressions for dr2 and the volume element
in paraboloidal co-ordinates.11.19 Elliptic co-ordinates in a plane are defined by
where α is a positive constant, with 0 ≤ u < ∞ and 0 ≤ w < 2π. Show that (u, w) are orthogonal co-ordinates and that the lines u = constant, w = constant correspond to an ellipse and a hyperbola, respectively. Take 0 < w < π/2, so that the point of intersection of these lines lies in the positive quadrant. Sketch the lines, and indicate the co-ordinate axes eu, ew at this point.
11.20 If F = xzi + xj − 2y2k, evaluate the integral
where Ω is the volume bounded by the surfaces x = 0, y = 0, y = 3, z = x2, z = 2.
11.21 Evaluate the integral
over the octant bounded by the co-ordinate planes x = 0, y = 0, z = 0 and the sphere x2 + y2 + z2 = a2.
[Hint: the integral
may be useful]
11.22 A container in the shape of a hemisphere of radius R is held so that its flat top is horizontal, and filled with liquid to a height h < R. What is the volume occupied by the liquid?
11.23 Using the result
evaluate the integral
over the tetrahedron bounded by the co-ordinate planes and the plane P: x + y + z = 1.
