Page:Calculus Made Easy.pdf/224

elements as $u\cdot dx\cdot dy$ (that is to say, of the value of $u$ over a little rectangle $dx$ long and $dy$ broad) has to be summed up over the whole length and whole breadth.

Similarly in the case of solids, where we deal with three dimensions. Consider any element of volume, the small cube whose dimensions are $dx\;dy\;dz$. If the figure of the solid be expressed by the function $f(x,y,z)$, then the whole solid will have the volume-integral,

${\text{volume}}=\iiint f(x,y,z)\cdot dx\cdot dy\cdot dz$.

Naturally, such integrations have to be taken between appropriate limits ^[1] in each dimension; and the integration cannot be performed unless one knows in what way the boundaries of the surface depend on $x$, $y$, and $z$. If the limits for $x$ are from $x_{1}$ to $x_{2}$, those for $y$ from $y_{1}$ to $y_{2}$, and those for $z$ from $z_{1}$ to $z_{2}$, then clearly we have

${\text{volume}}=\int _{z1}^{z2}\int _{y1}^{y2}\int _{x1}^{x2}f(x,y,z)\cdot dx\cdot dy\cdot dz$.

There are of course plenty of complicated and difficult cases; but, in general, it is quite easy to see the significance of the symbols where they are intended to indicate that a certain integration has to be performed over a given surface, or throughout a given solid space.

1. See p. 208 for integration between limits.