Triple Integrals

of two variables over a rectangular region in the plane. In this section we define the triple integral of a function

f (x, y, z)

Later in this section we extend the definition to more general regions in

ℝ^{3} .

Integrable Functions of Three Variables

subboxes

B_{i j k} = [x_{i - 1}, x_{i}] \times [y_{i - 1}, y_{i}] \times [z_{i - 1}, z_{i}],

We define the triple integral in terms of the limit of a triple Riemann sum, as we did for the double integral in terms of a double Riemann sum.

Therefore, we will use continuous functions for our examples. However, continuity is sufficient but not necessary; in other words,

f

and all the properties of a double integral apply to a triple integral. Just as the double integral has many practical applications, the triple integral also has many applications, which we discuss in later sections.

Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as in the case of the double integral, we can have an iterated triple integral, and consequently, a version of Fubini’s thereom for triple integrals exists.

real numbers, the iterated triple integral can be expressed in six different orderings:

For a rectangular box, the order of integration does not make any significant difference in the level of difficulty in computation. We compute triple integrals using Fubini’s Theorem rather than using the Riemann sum definition. We follow the order of integration in the same way as we did for double integrals (that is, from inside to outside).

Triple Integrals over a General Bounded Region

We now expand the definition of the triple integral to compute a triple integral over a more general bounded region

E

Changing the Order of Integration

As we have already seen in double integrals over general bounded regions, changing the order of the integration is done quite often to simplify the computation. With a triple integral over a rectangular box, the order of integration does not change the level of difficulty of the calculation. However, with a triple integral over a general bounded region, choosing an appropriate order of integration can simplify the computation quite a bit. Sometimes making the change to polar coordinates can also be very helpful. We demonstrate two examples here.

Average Value of a Function of Three Variables

Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.

Key Concepts

and is also equal to any of the other five possible orderings for the iterated triple integral.

Key Equations

In the following exercises, evaluate the triple integrals over the rectangular solid box $B .$

\underset{B}{∭} (2 x + 3 y^{2} + 4 z^{3}) d V,

where $B = {(x, y, z) \| 0 \leq x \leq 1, 0 \leq y \leq 2, 0 \leq z \leq 3}$

192

\underset{B}{∭} (x y + y z + x z) d V,

where $B = {(x, y, z) \| 1 \leq x \leq 2, 0 \leq y \leq 2, 1 \leq z \leq 3}$

\underset{B}{∭} (x cos y + z) d V,

where $B = {(x, y, z) \| 0 \leq x \leq 1, 0 \leq y \leq π, −1 \leq z \leq 1}$

0

\underset{B}{∭} (z sin x + y^{2}) d V,

where $B = {(x, y, z) \| 0 \leq x \leq π, 0 \leq y \leq 1, −1 \leq z \leq 2}$

In the following exercises, change the order of integration by integrating first with respect to $z,$

then $x,$

then $y .$

\int_{0}^{1} \int_{1}^{2} \int_{2}^{3} (x^{2} + ln y + z) d x d y d z

\int_{1}^{2} \int_{2}^{3} \int_{0}^{1} (x^{2} + ln y + z) d z d x d y = \frac{35}{6} + 2 ln 2

\int_{0}^{1} \int_{−1}^{1} \int_{0}^{3} (z e^{x} + 2 y) d x d y d z

\int_{−1}^{2} \int_{1}^{3} \int_{0}^{4} (x^{2} z + \frac{1}{y}) d x d y d z

\int_{1}^{3} \int_{0}^{4} \int_{−1}^{2} (x^{2} z + \frac{1}{y}) d z d x d y = 64 + 12 ln 3

\int_{1}^{2} \int_{−2}^{−1} \int_{0}^{1} \frac{x + y}{z} d x d y d z

Let $F, G, and H$

be continuous functions on $[a, b], [c, d],$

and $[e, f],$

respectively, where $a, b, c, d, e, and f$

are real numbers such that $a < b, c < d, and e < f .$

Show that

\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} F (x) G (y) H (z) d z d y d x = (\int_{a}^{b} F (x) d x) (\int_{c}^{d} G (y) d y) (\int_{e}^{f} H (z) d z) .

Let $F, G, and H$

be differential functions on $[a, b], [c, d],$

and $[e, f],$

respectively, where $a, b, c, d, e, and f$

are real numbers such that $a < b, c < d, and e < f .$

Show that

\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} F^{'} (x) G^{'} (y) H^{'} (z) d z d y d x = [F (b) - F (a)] [G (d) - G (c)] [H (f) - H (e)] .

In the following exercises, evaluate the triple integrals over the bounded region $E = {(x, y, z) \| a \leq x \leq b, h_{1} (x) \leq y \leq h_{2} (x), e \leq z \leq f} .$

\underset{E}{∭} (2 x + 5 y + 7 z) d V,

where $E = {(x, y, z) \| 0 \leq x \leq 1, 0 \leq y \leq - x + 1, 1 \leq z \leq 2}$

\frac{77}{12}

\underset{E}{∭} (y ln x + z) d V,

where $E = {(x, y, z) \| 1 \leq x \leq e, 0 \leq y \leq ln x, 0 \leq z \leq 1}$

\underset{E}{∭} (sin x + sin y) d V,

where $E = {(x, y, z) \| 0 \leq x \leq \frac{π}{2}, − cos x \leq y \leq cos x, −1 \leq z \leq 1}$

2

\underset{E}{∭} (x y + y z + x z) d V,

where $E = {(x, y, z) \| 0 \leq x \leq 1, − x^{2} \leq y \leq x^{2}, 0 \leq z \leq 1}$

In the following exercises, evaluate the triple integrals over the indicated bounded region $E .$

\underset{E}{∭} (x + 2 y z) d V,

where $E = {(x, y, z) \| 0 \leq x \leq 1, 0 \leq y \leq x, 0 \leq z \leq 5 - x - y}$

\frac{439}{120}

\underset{E}{∭} (x^{3} + y^{3} + z^{3}) d V,

where $E = {(x, y, z) \| 0 \leq x \leq 2, 0 \leq y \leq 2 x, 0 \leq z \leq 4 - x - y}$

\underset{E}{∭} y d V,

where $E = {(x, y, z) \| - 1 \leq x \leq 1, − \sqrt{1 - x^{2}} \leq y \leq \sqrt{1 - x^{2}}, 0 \leq z \leq 1 - x^{2} - y^{2}}$

0

\underset{E}{∭} x d V,

where $E = {(x, y, z) \| - 2 \leq x \leq 2, −4 \sqrt{1 - x^{2}} \leq y \leq \sqrt{4 - x^{2}}, 0 \leq z \leq 4 - x^{2} - y^{2}}$

In the following exercises, evaluate the triple integrals over the bounded region $E$

of the form $E = {(x, y, z) \| g_{1} (y) \leq x \leq g_{2} (y), c \leq y \leq d, e \leq z \leq f} .$

\underset{E}{∭} x^{2} d V,

where $E = {(x, y, z) \| 1 - y^{2} \leq x \leq y^{2} - 1, −1 \leq y \leq 1, 1 \leq z \leq 2}$

- \frac{64}{105}

\underset{E}{∭} (sin x + y) d V,

where $E = {(x, y, z) \| - y^{4} \leq x \leq y^{4}, 0 \leq y \leq 2, 0 \leq z \leq 4}$

\underset{E}{∭} (x - y z) d V,

where $E = {(x, y, z) \| - y^{6} \leq x \leq \sqrt{y}, 0 \leq y \leq 1 x, −1 \leq z \leq 1}$

\frac{11}{26}

\underset{E}{∭} z d V,

where $E = {(x, y, z) \| 2 - 2 y \leq x \leq 2 + \sqrt{y}, 0 \leq y \leq 1 x, 2 \leq z \leq 3}$

In the following exercises, evaluate the triple integrals over the bounded region

E = {(x, y, z) \| g_{1} (y) \leq x \leq g_{2} (y), c \leq y \leq d, u_{1} (x, y) \leq z \leq u_{2} (x, y)} .

\underset{E}{∭} z d V,

where $E = {(x, y, z) \| - y \leq x \leq y, 0 \leq y \leq 1, 0 \leq z \leq 1 - x^{4} - y^{4}}$

\frac{113}{450}

\underset{E}{∭} (x z + 1) d V,

where $E = {(x, y, z) \| 0 \leq x \leq \sqrt{y}, 0 \leq y \leq 2, 0 \leq z \leq 1 - x^{2} - y^{2}}$

\underset{E}{∭} (x - z) d V,

where $E = {(x, y, z) \| - \sqrt{1 - y^{2}} \leq x \leq y, 0 \leq y \leq \frac{1}{2} x, 0 \leq z \leq 1 - x^{2} - y^{2}}$

\frac{1}{160} (6 \sqrt{3} - 41)

\underset{E}{∭} (x + y) d V,

where $E = {(x, y, z) \| 0 \leq x \leq \sqrt{1 - y^{2}}, 0 \leq y \leq 1 x, 0 \leq z \leq 1 - x}$

In the following exercises, evaluate the triple integrals over the bounded region

E = {(x, y, z) \| (x, y) \in D, u_{1} (x, y) x \leq z \leq u_{2} (x, y)},

where $D$

is the projection of $E$

onto the $x y$

-plane.

\iint_{D} (\int_{1}^{2} (x + z) d z) d A,

where $D = {(x, y) \| x^{2} + y^{2} \leq 1}$

\frac{3 π}{2}

\iint_{D} (\int_{1}^{3} x (z + 1) d z) d A,

where $D = {(x, y) \| x^{2} - y^{2} \geq 1, x \leq \sqrt{5}}$

\iint_{D} (\int_{0}^{10 - x - y} (x + 2 z) d z) d A,

where $D = {(x, y) \| y \geq 0, x \geq 0, x + y \leq 10}$

1250

\iint_{D} (\int_{0}^{4 x^{2} + 4 y^{2}} y d z) d A,

where $D = {(x, y) \| x^{2} + y^{2} \leq 4, y \geq 1, x \geq 0}$

The solid $E$

bounded by $y^{2} + z^{2} = 9, z = 0,$

and $x = 5$

is shown in the following figure. Evaluate the integral $\underset{E}{∭} z d V$

by integrating first with respect to $z,$

then $y, and then x .$

A solid arching shape that reaches its maximum along the y axis with z = 3. The shape reach zero at y = plus or minus 3, and the graph is truncated at x = 0 and 5.

\int_{0}^{5} \int_{−3}^{3} \int_{0}^{\sqrt{9 - y^{2}}} z d z d y d x = 90

The solid $E$

bounded by $y = \sqrt{x},$

x = 4,

y = 0,

and $z = 1$

is given in the following figure. Evaluate the integral $\underset{E}{∭} x y z d V$

by integrating first with respect to $x,$

then $y,$

and then $z .$

A quarter section of an oval cylinder with z from negative 2 to positive 1. The solid is bounded by y = 0 and x = 4, and the top of the shape runs from (0, 0, 1) to (4, 2, 1) in a gentle arc.

[T] The volume of a solid $E$

is given by the integral $\int_{−2}^{0} \int_{x}^{0} \int_{0}^{x^{2} + y^{2}} d z d y d x .$

Use a computer algebra system (CAS) to graph $E$

and find its volume. Round your answer to two decimal places.

V = 5.33

A complex shape that starts at the origin and reaches its maximum at (negative 2, negative 2, 8). The shape is truncated by the x = y plane, the x = 0 plane, the y = negative 2 plane, the z = 0 plane, and a complex triangular-like shape with curved edges and sides (negative 2, negative 2, 8), (0, 0, 0), and (0, negative 2, 4).

[T] The volume of a solid $E$

is given by the integral $\int_{−1}^{0} \int_{− x^{2}}^{0} \int_{0}^{1 + \sqrt{x^{2} + y^{2}}} d z d y d x .$

Use a CAS to graph $E$

and find its volume $V .$

Round your answer to two decimal places.

In the following exercises, use two circular permutations of the variables $x, y, and z$

to write new integrals whose values equal the value of the original integral. A circular permutation of $x, y, and z$

is the arrangement of the numbers in one of the following orders: $y, z, and x or z, x, and y .$

\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (x^{2} z^{2} + 1) d x d y d z

\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (y^{2} z^{2} + 1) d z d x d y;

\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (x^{2} y^{2} + 1) d y d z d x

\int_{1}^{3} \int_{0}^{1} \int_{0}^{− x + 1} (2 x + 5 y + 7 z) d y d x d z

\int_{0}^{1} \int_{− y}^{y} \int_{0}^{1 - x^{4} - y^{4}} ln x d z d x d y

\int_{−1}^{1} \int_{0}^{1} \int_{− y^{6}}^{\sqrt{y}} (x + y z) d x d y d z

Set up the integral that gives the volume of the solid $E$

bounded by $y^{2} = x^{2} + z^{2}$

and $y = a^{2},$

where $a > 0 .$

V = \int_{− a}^{a} \int_{− \sqrt{a^{2} - z^{2}}}^{\sqrt{a^{2} - z^{2}}} \int_{\sqrt{x^{2} + z^{2}}}^{a^{2}} d y d x d z

Set up the integral that gives the volume of the solid $E$

bounded by $x = y^{2} + z^{2}$

and $x = a^{2},$

where $a > 0 .$

Find the average value of the function $f (x, y, z) = x + y + z$

over the parallelepiped determined by $x = 0, x = 1, y = 0, y = 3, z = 0,$

and $z = 5 .$

\frac{9}{2}

Find the average value of the function $f (x, y, z) = x y z$

over the solid $E = [0, 1] \times [0, 1] \times [0, 1]$

situated in the first octant.

Find the volume of the solid $E$

that lies under the plane $x + y + z = 9$

and whose projection onto the $x y$

-plane is bounded by $x = \sqrt{y - 1}, x = 0,$

and $x + y = 7 .$

\frac{156}{5}

Find the volume of the solid E that lies under the plane $2 x + y + z = 8$

and whose projection onto the $x y$

-plane is bounded by $x = {sin}^{−1} y, y = 0,$

and $x = \frac{π}{2} .$

Consider the pyramid with the base in the $x y$

-plane of $[−2, 2] \times [−2, 2]$

and the vertex at the point $(0, 0, 8) .$

Show that the equations of the planes of the lateral faces of the pyramid are $4 y + z = 8,$ $4 y - z = −8,$ $4 x + z = 8,$
and
$−4 x + z = 8 .$
Find the volume of the pyramid.

a. Answers may vary; b. $\frac{128}{3}$

Consider the pyramid with the base in the $x y$

-plane of $[−3, 3] \times [−3, 3]$

and the vertex at the point $(0, 0, 9) .$

Show that the equations of the planes of the side faces of the pyramid are $3 y + z = 9,$ $3 y + z = 9,$ $y = 0$
and
$x = 0 .$
Find the volume of the pyramid.

The solid $E$

bounded by the sphere of equation $x^{2} + y^{2} + z^{2} = r^{2}$

with $r > 0$

and located in the first octant is represented in the following figure.

The eighth of a sphere of radius 2 with center at the origin for positive x, y, and z.

Write the triple integral that gives the volume of $E$
by integrating first with respect to
$z,$
then with
$y,$
and then with
$x .$
Rewrite the integral in part a. as an equivalent integral in five other orders.

a. $\int_{0}^{4} \int_{0}^{\sqrt{r^{2} - x^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}} d z d y d x;$

b. $\int_{0}^{2} \int_{0}^{\sqrt{r^{2} - y^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}} d z d x d y,$

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - z^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}} d y d x d z,

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - x^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}} d y d z d x,

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - z^{2}}} \int_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}} d x d y d z,

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - y^{2}}} \int_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}} d x d z d y

The solid $E$

bounded by the equation $9 x^{2} + 4 y^{2} + z^{2} = 1$

and located in the first octant is represented in the following figure.

In the first octant, a complex shape is shown that is roughly a solid ovoid with center the origin, height 1, width 0.5, and length 0.35.

Write the triple integral that gives the volume of $E$
by integrating first with respect to
$z,$
then with
$y,$
and then with
$x .$
Rewrite the integral in part a. as an equivalent integral in five other orders.

Find the volume of the prism with vertices $(0, 0, 0), (2, 0, 0), (2, 3, 0),$

(0, 3, 0), (0, 0, 1), and (2, 0, 1) .

3

Find the volume of the prism with vertices $(0, 0, 0), (4, 0, 0), (4, 6, 0),$

(0, 6, 0), (0, 0, 1), and (4, 0, 1) .

The solid $E$

bounded by $z = 10 - 2 x - y$

and situated in the first octant is given in the following figure. Find the volume of the solid.

A tetrahedron bounded by the x y, y z, and x z planes and a triangle with vertices (0, 0, 10), (5, 0, 0), and (0, 10, 0).

\frac{250}{3}

The solid $E$

bounded by $z = 1 - x^{2}$

and situated in the first octant is given in the following figure. Find the volume of the solid.

A complex shape in the first octant with height 1, width 5, and length 1. The shape appears to be a slightly deformed quarter of a cylinder of radius 1 and width 5.

The midpoint rule for the triple integral $\underset{B}{∭} f (x, y, z) d V$

over the rectangular solid box $B$

is a generalization of the midpoint rule for double integrals. The region $B$

is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum $\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{n} f (\bar{x_{i}}, \bar{y_{j}}, \bar{z_{k}}) Δ V,$

where $(\bar{x_{i}}, \bar{y_{j}}, \bar{z_{k}})$

is the center of the box $B_{i j k}$

and $Δ V$

is the volume of each subbox. Apply the midpoint rule to approximate $\underset{B}{∭} x^{2} d V$

over the solid $B = {(x, y, z) \| 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1}$

by using a partition of eight cubes of equal size. Round your answer to three decimal places.

\frac{5}{16} \approx 0.313

[T]

Apply the midpoint rule to approximate $\underset{B}{∭} e^{− x^{2}} d V$
over the solid
$B = {(x, y, z) \| 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1}$
by using a partition of eight cubes of equal size. Round your answer to three decimal places.
Use a CAS to improve the above integral approximation in the case of a partition of $n^{3}$
cubes of equal size, where
$n = 3, 4,…, 10 .$

Suppose that the temperature in degrees Celsius at a point $(x, y, z)$

of a solid $E$

bounded by the coordinate planes and $x + y + z = 5$

is $T (x, y, z) = x z + 5 z + 10 .$

Find the average temperature over the solid.

\frac{35}{2}

Suppose that the temperature in degrees Fahrenheit at a point $(x, y, z)$

of a solid $E$

bounded by the coordinate planes and $x + y + z = 5$

is $T (x, y, z) = x + y + x y .$

Find the average temperature over the solid.

Show that the volume of a right square pyramid of height $h$

and side length $a$

is $v = \frac{h a^{2}}{3}$

by using triple integrals.

Show that the volume of a regular right hexagonal prism of edge length $a$

is $\frac{3 a^{3} \sqrt{3}}{2}$

by using triple integrals.

Show that the volume of a regular right hexagonal pyramid of edge length $a$

is $\frac{a^{3} \sqrt{3}}{2}$

by using triple integrals.

If the charge density at an arbitrary point $(x, y, z)$

of a solid $E$

is given by the function $ρ (x, y, z),$

then the total charge inside the solid is defined as the triple integral $\underset{E}{∭} ρ (x, y, z) d V .$

Assume that the charge density of the solid $E$

enclosed by the paraboloids $x = 5 - y^{2} - z^{2}$

and $x = y^{2} + z^{2} - 5$

is equal to the distance from an arbitrary point of $E$

to the origin. Set up the integral that gives the total charge inside the solid $E .$

Integrable Functions of Three Variables