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Multiple Integral Definitions

April 13th, 2009 |

Author: Merrin

Multiple integrals are useful for making a mathematical connection with many multidimensional physical phenomena. A double multiple integral may looks like this

$\displaystyle \int _{a}^{b}\left[\displaystyle \int _{c}^{d}f(x,y)dy \right] dx$

The placement of dx to the left of the brackets is also acceptable.

$\displaystyle \int _{a}^{b}\left[\displaystyle \int _{c}^{d}f(x,y)dy \right] dx=\displaystyle \int _{a}^{b}dx\left[\displaystyle \int _{c}^{d}f(x,y)dy \right]$

It is not a giant leap to evaluate a double integral, just calculate the quantity in the brackets as if it were a normal single integral with x treated as a constant. The result immediately plugs into the location for the integrand of the second integral then do the last one.

A triple integral looks like this

$\displaystyle \int _{x_{1} }^{x_{2} }\left[\displaystyle \int _{y_{1} }^{y_{2} }\left[\displaystyle \int _{z_{1} }^{z_{2} }f(x,y,z)dz \right] dy\right] dx$

In practice, it is customary to drop the brackets, but they clarify how the integration is supposed to be carried out. Just like for writing single integrals with variable limits of integration, the same is possible for multiple integrals.

Variable limits of integration in a double integral look like this

$\displaystyle \int _{a}^{b}\left[\displaystyle \int _{g(x)}^{h(x)}f(x,y)dy \right] dx$

In the case of multiple integrals, the variable limits can feed into the calculation of the next level of integration.

Definite integrals in one dimension represent the area under the curve. Double integrals the volume under a surface. This can be understood from the Riemann definitions of the respective integrals. The Riemann definition of the single definite integral over the interval $latex x\in [a,b]$

$\displaystyle \int _{a}^{b}f(x)dx=\mathop{\lim }\limits_{N\to \infty } \sum _{k=0}^{N-1}f(a+k\Delta x)\Delta x \quad \Delta x=\cfrac{b-a}{N}$

For double integration over the rectangular region $latex x\in [a,b], y\in [c,d]$

$\int _{a}^{b}\displaystyle \int _{c}^{d}f(x,y) dydx=\mathop{\lim }\limits_{N\to \infty } \mathop{\lim }\limits_{M\to \infty } \sum _{i=0}^{N-1}\sum _{j=0}^{M-1}f(a+i\Delta x,c+j\Delta y)\Delta x \Delta y$

${\Delta x=\cfrac{b-a}{N} \qquad \Delta y=\cfrac{d-c}{M} }$

This definition indicates a sum of parallelpipes with volumes of $latex f(x_{i} ,y_{i} ) $ multiplied by base areas of $latex \Delta x\Delta y$. Visualize an array of these parallelpipes to realize that a double integral of a function is actually a volume.

We can also sum up the volume another way by rearranging the double sum as a single sum of individual area elements.

$\displaystyle \iint \nolimits _{R}f(x,y)dydx =\mathop{\lim }\limits_{N\to \infty } \sum _{p=1}^{N}f(x_{p} ,y_{p} )\Delta A_{p}$

Besides integrating over a rectangle, we will want to integrate over a more general region. We can introduce a function $latex \theta _{ij}$ which is 1 inside the region of integration and 0 outside of it and multiply it into the sum. This way we can pick the points in the plane that should count for the region of integration. The limits of a double integral can specify simple region of integration.

The Riemann definition for an arbitrary volume is not much different

$\displaystyle \int\!\!\displaystyle \int\!\!\displaystyle \int _{V}f(x,y,z)dzdydx=\mathop{\lim }\limits_{N\to \infty } \sum _{p=1}^{N}f(x_{p} ,y_{p} ,z_{p} )\Delta V_{p}$

The geometric interpretation of a triple integral is a hypervolume, but there are other simpler interpretations. Imagine the integrand is mass density for example. The integral of the mass density over a volume gives the mass, a useful physical property.

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