Calculus/Multiple integration

Double integration
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 $$\mathbb R^2,\mathbb R^3$$ or $$\mathbb R^n$$ in general. (Actually, we only have the term interval in $$\mathbb R$$.)

In multivariable case, we need to consider not just 'interval' itself (which is undefined in multivariable case), but Cartesian product over intervals for $$\mathbb R^2$$, and more generally n-ary Cartesian product over intervals for $$\mathbb R^n$$.

Area (for $$n=2$$), volume (for $$n=3$$) or measure (for each positive number $$n$$) of geometric objects (e.g. rectangles in $$\mathbb R^2$$ and cubes in $$\mathbb R^3$$) in $$\mathbb R^n$$ is the product of the lengths of all its sides (in different dimensions).

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.

A physical meaning of double integration is computing volume.

Let's also introduce some properties of double integral to ease computation of double integral.

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 iterated integral, which is used in the results.

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.

{ Example. }

{Choose the correct expression(s) for the integral $$\iint_{[1,3]\times[2,4]}x^2y\,dx\,dy$$. + $$\int_1^3\int_2^4x^2y\,dy\,dx$$ + $$\int_2^4\int_1^3x^2y\,dx\,dy$$ - $$\int_2^4\int_1^3x^2y\,dy\,dx$$ - $$\int_1^3\int_2^4x^2y\,dx\,dy$$
 * type="[]"}
 * This is correct by Fubini's theorem.
 * This is correct by Fubini's theorem.
 * Be careful about the change of bounds.
 * Be careful about the change of bounds.

{Choose the correct expression(s) for the integral $$\iint_{[-1,1]\times [3,7]}y^2e^x\,dx\,dy$$. + $$\int_{-1}^1\left(e^7-e^3\right)y^2\,dy$$ - $$\int_{-1}^1 \frac{316}{3}e^x\,dy$$ - $$\int_{3}^7\left(e^7-e^3\right)y^2\,dx$$ - $$\int_{3}^7\left(e^7-e^3\right)y^2\,dy$$ + $$\int_{-1}^1 \frac{316}{3}e^x\,dx$$
 * type="[]"}

Double integrals over more general regions in R2
We have defined double integrals over rectangles in $$\mathbb R^2$$. 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 $$f:D\to\mathbb R$$ in which $$D\subseteq \mathbb R^2$$ 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 $$R$$ containing $$D$$ (i.e., $$R\supseteq D$$), and let $$f(x,y)=0$$ for each $$(x,y)\in R$$ lying outside $$D$$ (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 does not 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.

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.

Triple integration
The concepts in the section of double integrals apply to triple integrals (and also multiple integrals generally) analogously. We will give several examples for triple integrals in this section.