calculuspowerup.com

Calculating Multiple Integrals

April 12th, 2009 |

Author: Merrin

In practice, double integrals occur in two forms, constant limits or variable
limits.

$\displaystyle I_{1} =\int _{a}^{b}\int _{c}^{d}f(x,y)dydx\quad \quad \, I_{2} = \int _{a}^{b}\int _{\phi (x)}^{\chi(x)}f(x,y)dydx$

Expect to do about twice as much work as for a single integral. Double integrals with constant limits are straightforward to calculate. The calculation is an iterative procedure. By iterative, I mean ignore everything outside of the brackets for the moment and do the integral inside them. Repeat this process until the multiple integral is done. That stuff comes into play in the next step. In double integrals, requires treating some variables as constants, but don’t worry they all get integrated out eventually.

Example 1. Evaluate this double integral having constant limits of integration

$\displaystyle \int _{-1}^{1}\int _{0}^{1}x^{2} ydydx =\int _{-1}^{1}\left[\int _{0}^{1}x^{2} ydy \right]dx$

Solution 1. Calculate the integral in brackets as a single integral first treating x as a constant.

$\displaystyle \int _{-1}^{1}\left[\int _{0}^{1}x^{2} ydy \right]dx =\int _{-1}^{1}\left[\cfrac{1}{2} x^{2} y^{2} \right]_{y=0}^{y=1} dx$

$\displaystyle =\int _{-1}^{1}\cfrac{1}{2}x^{2} dx=\int _{0}^{1}x^{2}dx=\cfrac{1}{3}$

The other thing that can come up when calculating double integrals are variable limits of integration. I will tweak the last example a little bit so we can practice variable limits of integration. The calculation is not too difficult with variable limits. Plug in the functions to the spots where the constant limits normally are.

Example 2. Calculate the following double integral with variable limits.

$\displaystyle \int _{-1}^{1}\left[\int _{0}^{x^{2} }x^{2} ydy \right]dx$

Solution 2. This example is entirely the same except one of the limits is variable. This variable limit feeds into the final integration.

$\displaystyle \int _{-1}^{1}\left[\int _{0}^{x^{2} }x^{2} ydy \right]dx =\int _{-1}^{1}\left[\cfrac{1}{2} x^{2} y\left. ^{2} \right|_{y=0}^{y=x^{2} } \right]dx=$

$\displaystyle \int _{-1}^{1}\cfrac{1}{2} x^{4} dx=\int _{0}^{1}x^{4} dx =\cfrac{1}{5}$

Multiple integrals are not necessarily harder than regular integrals. They just require multiple integration steps using the normal techniques. Multiple integrals are always of the definite variety. Integration constants would screw everything up in multiple integrals because the constants would be actual functions of the other variables at each step.

We will learn a variety of techniques for calculating multiple integrals such as interchanging the order of integration, using separation of variables, transformations to more suitable coordinate systems, and using symmetry.

Back to Home:

Back to Index:

Previous Topic: Multiple Integral Definitions

Next Topic: Separable Integrals

Bookmark It