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Volume under surface

Double integral as volume under the surface z=10-\frac{x^2-y^2}{8} . The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.

A multiple integral is a definite integral of a function of more than one variable. A double integral gives the volume under a surface, and higher numbers of integrals give hypervolume, or the volume of an object in more than three dimensions. In theory, there is no limit to the number of integrals and variables, but in practice, a triple integral is usually the highest number of integrals with any practical purpose.

The domain of the integral is often denoted by a single letter; for example:

\iint_A f(x,y)\,dx\,dy

The domain does not necessarily have to be a constant. Functions can be used as the bounds as well. For example,

\int\limits_a^b\int\limits_{h(x)}^{g(x)}f(x,y)\,dy\,dx

or

\int\limits_a^b\int\limits_{h(y)}^{g(y)}f(x,y)\,dx\,dy

Multiple integrals can be solved by integrating for one variable at a time while treating the others as constants. For instance, to find the volume under the function z=2 in the region 0\le x\le2\ ,\ 0\le y\le2 , we would integrate as follows:

\int\limits_0^2\int\limits_0^2 2\,dy\,dx=\int\limits_0^2\Big[2y\Big]_0^2\,dx=\int\limits_0^2 4\,dx=\Big[4x\Big]_0^2=8

The order in which the variables are integrated does not matter. Another example:

Area under z-2y-3x^2=9 in the region 0\le x\le1\ ,\ 0\le y\le3

z=9+2y+x^2
\int\limits_0^3\int\limits_0^1(9-2y-3x^2)\,dx\,dy=\int\limits_0^3\Big[9x+2yx+x^3\Big]_0^1\,dy=\int\limits_0^3(9+2y+1)dy
=\Big[9y+y^2+y\Big]^3_0=27+9+3=39

Applications

Triple integrals are useful for calculating the net flux of a vector field with the divergence theorem. They are also used to calculate surface integrals.

Double integrals can also be used to find area (and triple integrals to find volume), since \iint_D dA=D and \iiint_D dV=D . This is often done by changing coordinates by using a Jacobian matrix.

A substance with the mass or charge density given by \rho(x,y,z) with the boundaries given by V has a mass or charge of \iiint_V \rho(x,y,z)dV .

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