Calculus/Multiple integration

Double integration

Illustration of partitioning interval

For (Riemann) integrals, we consider the Riemann sum. Recall in the one-variable case, we partition an interval into more and more subintervals with smaller and smaller width, and we are integrating over the interval by summing the areas of corresponding rectangles for each subinterval. For the multivariable case, we need to do something similar, but the problem arises when we need to partition 'interval' in ${ℝ}^2,{ℝ}^3$ or ${ℝ}^n$ in general. (Actually, we only have the term Template:Font color in $ℝ$.)

In multivariable case, we need to consider not just 'interval' itself (which is undefined in multivariable case), but Template:Font color over intervals for ${ℝ}^2$, and more generally Template:Font color over intervals for ${ℝ}^n$.

Illustration of Cartesian product by an example.

Template:Calculus/Def Template:Calculus/Def Area (for $n=2$), volume (for $n=3$) or measure (for each positive number $n$) of geometric objects (e.g. rectangles in ${ℝ}^2$ and cubes in ${ℝ}^3$) in ${ℝ}^n$ is the product of the lengths of all its sides (in different dimensions). Template:Calculus/Def Now, we are ready to define multiple integral in an analogous way compared with single integral. For simplicity, let us first discuss double integral, and then generalize it to multiple integral in an analogous way.

Illustration of partitioning rectangle in ${ℝ}^2$.

Template:Anchor Template:Calculus/Def Template:Calculus/Def A physical meaning of double integration is computing volume. Template:Anchor Template:Calculus/Def Template:Calculus/Def Let's also introduce some properties of double integral to ease computation of double integral. Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def

Iterated integrals

Thankfully, we need not always work with Riemann sums every time we want to calculate an integral in more than one variable. There are some results that make life a bit easier for us. Before stating the result, we need to define Template:Font color, which is used in the results. Template:Calculus/Def Template:Calculus/Def Computation of iterated integrals is generally much easier than computing the double integral directly using Riemann sum. So, it will be nice if we have some relationships between iterated integral and double integral for us to compute double integral with the help of iterated integral. It is indeed the case and the following theorem is the bridge between iterated integral and double integral. Template:Anchor Template:Calculus/Def Template:Calculus/Def <quiz display=simple> { Template:Font color }

{Choose the correct expression(s) for the integral $\underset{[1,3]\times [2,4]}{\iint }{x}^{2}ydxdy$. |type="[]"} + $\underset{1}{\overset{3}{\int }}\underset{2}{\overset{4}{\int }}{x}^{2}ydydx$ || This is correct by Fubini's theorem. + $\underset{2}{\overset{4}{\int }}\underset{1}{\overset{3}{\int }}{x}^{2}ydxdy$ || This is correct by Fubini's theorem. - $\underset{2}{\overset{4}{\int }}\underset{1}{\overset{3}{\int }}{x}^{2}ydydx$ || Be careful about the change of bounds. - $\underset{1}{\overset{3}{\int }}\underset{2}{\overset{4}{\int }}{x}^{2}ydxdy$ || Be careful about the change of bounds.

{Choose the correct expression(s) for the integral $\underset{[-1,1]\times [3,7]}{\iint }{y}^2{e}^{x}dxdy$. |type="[]"} + $\underset{-1}{\overset{1}{\int }}(e^7-e^3)y^{2}dy$ - $\underset{-1}{\overset{1}{\int }}\frac{316}{3}{e}^{x}dy$ - $\underset{3}{\overset{7}{\int }}(e^7-e^3)y^{2}dx$ - $\underset{3}{\overset{7}{\int }}(e^7-e^3)y^{2}dy$ + $\underset{-1}{\overset{1}{\int }}\frac{316}{3}{e}^{x}dx$ </quiz>

Double integrals over more general regions in R²

We have defined double integrals over rectangles in ${ℝ}^2$. However, we often want to compute double integral over regions with shape other than rectangle, e.g. circle, triangle, etc. Therefore, we will discuss an approach to compute double integrals over more general regions reasonably, without altering the definition of double integrals.

Consider a function $f:D\to ℝ$ in which $D\subseteq {ℝ}^2$ is a general region. To apply the definition of double integrals, we need to transform the general region $D$ to a rectangle (say $R$). An approach is finding a rectangle $R$ Template:Font color $D$ (i.e., $R\supseteq D$), and let $f(x,y)=0$ for each $(x,y)\in R$ lying Template:Font color $D$ (i.e., for each $(x,y)\in R\setminus D$ ). Because the value of the function is zero outside the region we are integrating over, this Template:Font color change the volume under the graph of $f$ over $D$, so this way is a good way to define such double integrals. Let's define such double integrals formally in the following. Template:Anchor Template:Calculus/Def Template:Calculus/Def However, this way of computation (by computing Riemann sum) is generally very difficult, and usually we use a generalized version of Fubini's theorem to compute such integrals. It will be discussed in the following. Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Hide Template:Calculus/Def Template:Hide Template:Calculus/Def Template:Hide Template:Anchor Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Hide

Triple integration

The concepts in the section of double integrals apply to Template:Font color (and also multiple integrals generally) analogously. We will give several examples for triple integrals in this section. Template:Calculus/Def Template:Calculus/Def Template:Anchor Template:Calculus/Def Template:Calculus/Def Template:Calculus/Def Template:Anchor Template:Calculus/Def Template:Calculus/Def Template:Hide

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