Page:Calculus Made Easy.pdf/223

On Double and Triple Integrals.

In many cases it is necessary to integrate some expression for two or more variables contained in it; and in that case the sign of integration appears more than once. Thus,

$\iint f(x,y,)\,dx\,dy$

means that some function of the variables $x$ and $y$ has to be integrated for each. It does not matter in which order they are done. Thus, take the function $x^{2}+y^{2}$. Integrating it with respect to $x$ gives us:

$\int (x^{2}+y^{2})\,dx={\tfrac {1}{3}}x^{3}+xy^{2}$.

Now, integrate this with respect to $y$:

$\int ({\tfrac {1}{3}}x^{3}+xy^{2})\,dy={\tfrac {1}{3}}x^{3}y+{\tfrac {1}{3}}xy^{3}$,

to which of course a constant is to be added. If we had reversed the order of the operations, the result would have been the same.

In dealing with areas of surfaces and of solids, we have often to integrate both for length and breadth, and thus have integrals of the form

$\iint u\cdot dx\,dy$,

where $u$ is some property that depends, at each point, on $x$ and on $y$. This would then be called a surface-integral. It indicates that the value of all such