(A.84) |

is a surface integral. For instance, the volume of water in a lake of depth is

(A.85) |

To evaluate this integral, we must split the calculation into two ordinary integrals. The volume in the strip shown in Figure A.17 is

(A.86) |

Note that the limits and depend on . The total volume is the sum over all strips: that is,

(A.87) |

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

(A.88) |

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

For example, consider

(A.89) |

where is shown in Figure A.18. Suppose that we evaluate the integral first:

(A.90) |

Let us now evaluate the integral:

(A.91) |

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

(A.92) |

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

(A.93) |

where is a unit square. This integral can be written

(A.94) |

because the limits are both independent of the other variable.