Surface integrals

Let us take a surface $S$, which is not necessarily co-planar, and divide in up into (scalar) elements $\delta S_i$. Then

$$\int\!\int_S f(x,y,z) dS = \lim_{\delta S_i\rightarrow 0}\sum_i f(x,y,z) \delta S_i$$ (75)

is a surface integral. For instance, the volume of water in a lake of depth $D(x,y)$ is

$$V= \int\!\int D(x,y) dS.$$ (76)

To evaluate this integral we must split the calculation into two ordinary integrals. The volume in the strip shown in Fig. 14 is

$$\left[\int_{x_1}^{x_2} D(x,y) dx\right] dy.$$ (77)

Note that the limits $x_1$ and $x_2$ depend on $y$. The total volume is the sum over all strips:

$$V = \int_{y_1}^{y_2} dy\left[\int_{x_1(y)}^{x_2(y)} D(x,y) dx\right] \equiv \int\!\int_S D(x,y) dx dy.$$ (78)

Of course, the integral can be evaluated by taking the strips the other way around:

$$V = \int_{x_1}^{x_2} dx \int_{y_1(x)}^{y_2(x)} D(x,y) dy.$$ (79)

Interchanging the order of integration is a very powerful and useful trick. But great care must be taken when evaluating the limits.

Figure 14:

As an example, consider

$$\int\!\int_S x^2 y dx dy,$$ (80)

where $S$ is shown in Fig. 15. Suppose that we evaluate the $x$ integral first:

$$dy\left(\int_0^{1-y} x^2 y dx\right) = y dy\left[ \frac{x^3}{3}\right]^{1-y}_0 = \frac{y}{3} (1-y)^3 dy.$$ (81)

Let us now evaluate the $y$ integral:

$$\int_0^1 \left(\frac{y}{3} - y^2 + y^3- \frac{y^4}{3}\right)dy = \frac{1}{60}.$$ (82)

We can also evaluate the integral by interchanging the order of integration:

$$\int_0^1 x^2 dx \int_0^{1-x} y dy = \int_0^1\frac{x^2}{2} (1-x)^2 dx = \frac{1}{60}.$$ (83)

Figure 15:

In some cases, a surface integral is just the product of two separate integrals. For instance,

$$\int\int_S x^2 y dx dy$$ (84)

where $S$ is a unit square. This integral can be written

$$\int_0^1 dx \int_0^1 x^2 y dy = \left(\int_0^1 x^2 dx\ri... ...nt_0^1 y dy\right) = \frac{1}{3} \frac{1}{2} = \frac{1}{6},$$ (85)

since the limits are both independent of the other variable.

In general, when interchanging the order of integration, the most important part of the whole problem is getting the limits of integration right. The only foolproof way of doing this is to draw a diagram.