# Triple Integrals

## Introduction

Motivated by the concept of double integrals, we want to consider the integrals of functions of three variables. The domain of integration will now be a three-dimensional solid region.

## Triple integrals over rectangular boxes

### Integral sums

Assume that $$f$$ is a function defined on a rectangular box $$[a,b]\times [c,d]\times [e,g]$$ in the space $$\mathbb R^3$$.

Consider three sequences of points \begin{eqnarray*}&&a=x_0\leq x_1\leq x_2\leq\cdots\leq x_n=b,\newline &&c=y_0\leq y_1\leq y_2\leq\cdots\leq y_m=d,\mbox{ and}\newline &&e=z_0\leq z_1\leq z_2\leq\cdots\leq z_p=g\end{eqnarray*} for $$m,n,p\in\mathbb N$$. The box $$[a,b]\times[c,d]\times[e,g]$$ is the union of rectangles $$[x_i,x_{i+1}]\times[y_j,y_{j+1}]\times[z_k,z_{k+1}]$$ for each choice of $$i$$, $$j$$, and $$k$$ such that $$i\in\{0,1,\dots, n-1\}$$, $$j\in\{0,1,\dots, m-1\}$$, and $$k\in\{0,1,\dots, p-1\}$$. Denote these rectangles by $$R_1$$, $$R_2$$, $$\dots$$, $$R_{mnp}$$. Denote by $$A_i$$ the area of the rectangle $$R_i$$ and let $$c_i$$ be an arbitrary point from the rectangle $$R_i$$. The integral sum (also known as the Riemann sum) corresponding to the partition into rectangles $$R_1$$, $$\dots$$, $$R_{mnp}$$ with the points $$c_1$$, $$\dots$$, $$c_{mnp}$$ is defined as $S(f,R_1,\dots, R_{mnp}, c_1,\dots, c_{mnp})= f(c_1)\cdot A_1+ f(c_2)\cdot A_2+\cdots + f(c_{mnp})\cdot A_{mnp}.$ The diameter of partition is defined as $\delta(R_1, \dots, R_{mnp})=\max\{A_1,\dots ,A_{mnp}\}.$

### Definition of triple integrals over rectangular boxes

Consider the function $$f:[a,b]\times[c,d]\times[e,g]\to \mathbb R$$, and consider the partitions of the box $$[a,b]\times[c,d]\times[e,g]$$ into smaller boxes. We say that the function $$f$$ is integrable if the integral sums $$S(f,R_1, \dots, R_{mnp}, c_1, \dots, c_{mnp})$$ converge as the diameters of partitions converge to $$0$$. Here is the formal definition:

Definition: Triple integral over rectangular box The function $$f:[a,b]\times[c,d]\times[e,g]\to\mathbb R$$ is integrable and its integral over the box $$[a,b]\times[c,d]\times[e,g]$$ is equal to $$I\in\mathbb R$$ if for each $$\varepsilon > 0$$ there exists $$\delta > 0$$ such that for every partition $$\{R_1,\dots, R_{mnp}\}$$ of $$[a,b]\times[c,d]\times[e,f]$$ with $$\delta(R_1,\dots, R_{mnp}) < \delta$$ we have $\left|S(f,R_1,\dots, R_{mnp},c_1,\dots, c_{mnp})-I\right| < \varepsilon$ for every choice of $$c_1\in R_1$$, $$c_2\in R_2$$, $$\dots$$, $$c_{mnp}\in R_{mnp}$$.

The value $$I$$ from the previous definition is often denoted as $I=\iiint_{[a,b]\times[c,d]\times[e,g]} f(x,y,z)\,dxdydz \quad\quad\quad \mbox{or} \quad\quad\quad I=\iiint_{[a,b]\times[c,d]\times[e,g]} f(x,y,z)\,dV.$

## Iterated integrals and Fubini’s theorem

Assume that $$f:[a,b]\times[c,d]\times[e,g]\to\mathbb R$$ is an integrable function. For each fixed $$(x,y)\in[a,b]\times[c,d]$$ the function $$z\mapsto f(x,y,z)$$ is a function of one variable and we can talk about single integral $$\int_e^g f(x,y,z)\,dz$$. The result of integration will be a function in $$(x,y)$$ so we can talk about $$\int_a^b\int_c^d \left(\int_e^g f(x,y,z)\,dz\right)\,dydx$$. This is called the iterated integral.

Fubini’s theorem in three dimension states that the iterated integrals are equal to the triple integral defined above.

Theorem (Fubini) If $$f:[a,b]\times[c,d]\times[e,g]\to\mathbb R$$ is a continuous function, then \begin{eqnarray*}\iiint_{[a,b]\times[c,d]\times[e,g]} f(x,y,z)\,dxdydz&=&\int_a^b \int_c^d\int_e^g f(x,y,z)\,dzdydx \newline &=& \int_a^b\int_e^g \int_c^d f(x,y,z)\,dydzdx\newline &=& \int_c^d\int_a^b \int_e^g f(x,y,z)\,dzdxdy\newline &=& \int_c^d \int_e^g\int_a^b f(x,y,z)\,dxdzdy\newline &=& \int_e^g\int_a^b \int_c^d f(x,y,z)\,dydxdz \newline &=& \int_e^g\int_c^d\int_a^b f(x,y,z)\,dxdydz .\end{eqnarray*}

Remark. Fubini’s theorem holds also if instead of continuity of $$f$$ we assumed that $$f$$ is bounded on $$[a,b]\times[c,d]\times[e,g]$$, the set of discontinuities consists of finitely many smooth surfaces, and the iterated integrals exist.

We will omit the proof of the theorem.

## Triple integrals over general regions

Our goal now is to generalize the integration to include functions that are defined on regions that are not rectangular boxes.

Assume now that $$D\subseteq \mathbb R^3$$ is a general region that is not necessarily a rectangular box. We can define the integral $$\iint_D f(x,y,z)\,dV$$ in the following way. First, assume that $$R$$ is a rectangle of the form $$[a,b]\times[c,d]\times[e,g]$$ for some $$a,b,c,d,e,g\in\mathbb R$$. Then we define the function $$\hat f:R\to\mathbb R$$ in the following way $\hat f(x,y,z)=\left\{\begin{array}{ll} f(x,y,z),& \mbox{ if } (x,y,z)\in D,\newline 0,&\mbox{ if } (x,y,z)\not\in D,\end{array}\right.$ and we define $$\iiint_Df(x,y,z)\,dV$$ as: $\iint_Df(x,y,z)\,dV=\iiint_R\hat f(x,y,z)\,dV.$

## Practice problems

Problem 1. Evaluate the integral $\iiint_S xyz\,dxdydz,$ where $$S$$ is the rectangular box $$S=[0,2]\times[1,5]\times[-1,3]$$.

Problem 2. Evaluate the integral $\iiint_S z\,dxdydz,$ where $$S$$ is a triangular prism bounded by the planes $$x=0$$, $$y=0$$, $$z=0$$, $$x=5$$, and $$y+z=1$$.

Problem 3. Find the volume of the solid in the first octant that is located between the surface $$y=z^2$$ and the plane $$x+y=4$$.

Problem 4. Evaluate the integral $\int_{-1}^1\int_0^{z^2}\int_0^y \frac{z+\tan\frac{z}{x^2+y^2+1}}{x^2+y^2+1}\,dxdydz.$

Problem 5. For any bounded continuous function $$f$$ the integral $$\int_0^x\int_0^y\int_0^zf(t)\,dtdzdy$$ is equal to:
• (A) $$0$$
• (B) $$\int_0^xxf(t)\,dt$$
• (C) $$\frac12\int_0^x(x-t)^2f(t)\,dt$$
• (D) $$\int_0^x(y-x)f(y)dy$$
• (E) $$xf(x)-\frac{x^2}2$$