Multiple Integral

Evaluating Multiple Integrals

Multiple Integrals are evaluated from the inside out, beginning by evaluating the innermost integral, then working outwards.

${\begin{aligned}A&=\int \limits _{1}^{3}\int \limits _{0}^{x^{2}}\,dy\,dx\\&=\int \limits _{1}^{3}\left(\int \limits _{0}^{x^{2}}\,dy\right)\,dx\\&=\int \limits _{1}^{3}x^{2}\,dx\\A&={\frac {26}{3}}\\\end{aligned}}$

The inner integrals may have limits containing variables, so long as those variables are integrated in an enclosing integral. Because of this, the limits of outermost integrals must contain only constants.

Changing the Order of Integration

So long as the order of integration is changed correctly, the multiple integral will cover the same region, and therefore order will not affect the end result of the multiple integral. In general, it is wise to begin by establishing the limits of the outermost integral first, then working inwards, to avoid any mistakes.

Converting Coordinate Systems

Cartesian to Cylindrical

Cartesian to Spherical

Cylindrical to Spherical

Uses

Average Value

The Average value of a function $f(x)$ is equal to ${\frac {\iint \limits _{R}\,f(x)\,dA}{\iint \limits _{R}\,dA}}$

Areas/Volumes

The equation for Area is $\iint \limits _{R}\,dA$ and Volume is $\iiint \limits _{R}\,dV$

In Cartesian coordinates, $dA=dx\,dy$ and $dV=dx\,dy\,dz$ , therefore Area and Volume are $\iint \limits _{R}\,dx\,dy$ and $\iiint \limits _{R}\,dx\,dy\,dz$

The same process can be used in Polar, Cylindrical, and Spherical coordinates, as follows:

In Polar, $dA=r\,d\theta \,dr$

In Cylindrical, $dV=r\,d\theta \,dr\,dz$

In Spherical, $dV=\rho ^{2}\,\sin {(\phi )}\,d\rho \,d\phi \,d\theta$

Masses

The equation for the mass of an object is $\iiint \limits _{R}\,\sigma \,dV$ , where $\sigma$ is the density of the object (which could be either a constant or function of position)

Moments

First Moments

$\iiint \limits _{R}\,r\,\sigma \,dV$ , where r is the distance from the axis or line of rotation

Second Moments

$\iiint \limits _{R}\,r^{2}\,\sigma \,dV$ , where r is the distance from the axis or line of rotation

Center of Masses

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