Triple Integrals (Exercises 2)

Terms and Concepts

1. The strategy for establishing bounds for triple integrals is "from ________ to ________, then from ________ to ________ and then from ________ to ________."

Answer:

We integrate from surface to surface, then from curve to curve and then from point to point.

2. Give an informal interpretation of what $\int\int\int_Q \,dV$ means.

Answer:

$\int\int\int_Q \,dV$ = Volume of the solid region $Q$

3. Give two uses of triple integration.

Answer:

To compute total mass or average density of a solid object, given a density function or to compute the average temperature in a solid region or object.

4. If an object has a constant density $\delta$ and a volume $V$, what is its mass?

Answer:

It's mass is $\delta V$.

Volume of Solid Regions

In Exercises 5-8, two surfaces $f_1(x,y)$ and $f_2(x,y)$ and a region $R$ in the $xy$-plane are given. Set up and evaluate the triple integral that represents the volume between these surfaces over $R$.

5. $f_1(x,y) = 8-x^2-y^2,\,f_2(x,y) =2x+y;$
$R$ is the square with corners $(-1,-1)$ and $(1,1)$.

6. $f_1(x,y) = x^2+y^2,\,f_2(x,y) =-x^2-y^2;$
$R$ is the square with corners $(0,0)$ and $(2,3)$.

7. $f_1(x,y) = \sin x \cos y,\,f_2(x,y) =\cos x \sin y +2;$
$R$ is the triangle with corners $(0,0), \,(\pi , 0)$ and $(\pi,\pi)$.

8. $f_1(x,y) = 2x^2+2y^2+3,\,f_2(x,y) =6-x^2-y^2;$
$R$ is the circle $x^2+y^2=1$.

In Exercises 9-16, a domain $D$ is described by its bounding surfaces, along with a graph. Set up the triple integral that gives the volume of $D$ in the indicated order of integration, and evaluate the triple integral to find this volume.

9. $D$ is bounded by the coordinate planes and $z=2-\frac{2}{3}x-2y$.
Evaluate the triple integral with order $dz\,dy\,dx$.

10. $D$ is bounded by the planes $y=0,y=2,x=1,z=0$ and $z=(2-x)/2$.
Evaluate the triple integral with order $dx\,dy\,dz$.

11. $D$ is bounded by the planes $x=0,x=2,z=-y$ and by $z=y^2/2$.
Evaluate the triple integral with order $dy\,dz\,dx$.

12. $D$ is bounded by the planes $z=0,y=9, x=0$ and by $z=\sqrt{y^2-9x^2}$.
Do not evaluate any triple integral. Just set this one up.

13. $D$ is bounded by the planes $x=2,y=1,z=0$ and $z=2x+4y-4$.
Evaluate the triple integral with order $dx\,dy\,dz$.

14. $D$ is bounded by the plane $z=2y$ and by $y=4-x^2$.
Evaluate the triple integral with order $dz\,dy\,dx$.

15. $D$ is bounded by the coordinate planes and $y=1-x^2$ and $y=1-z^2$.
Do not evaluate any triple integral. Which order would be easier to evaluate: $dz\,dy\,dx$ or $dy\,dz\,dx$? Explain why.

16. $D$ is bounded by the coordinate planes and by $z=1-y/3$ and $z=1-x$.
Evaluate the triple integral with order $dx\,dy\,dz$.

In Exercises 17-20, evaluate the triple integral.

17. $\displaystyle \int_{-\pi/2}^{\pi/2}\int_{0}^{\pi}\int_{0}^{\pi} (\cos x \sin y \sin z )\,dz\,dy\,dx$

18. $\displaystyle \int_{0}^{1}\int_{0}^{x}\int_{0}^{x+y} (x+y+z )\,dz\,dy\,dx$

19. $\displaystyle \int_{0}^{\pi}\int_{0}^{1}\int_{0}^{z} (\sin (yz))\,dx\,dy\,dz$

20. $\displaystyle \int_{\pi}^{\pi^2}\int_{x}^{x^3}\int_{-y^2}^{y^2} (\cos x \sin y \sin z )\,dz\,dy\,dx$

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

$$\iiint_B (2x + 3y^2 + 4z^3) \space dV,$$ where $B = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\}$

[Hide Solution]

$192$

$$\iiint_B (xy + yz + xz) \space dV,$$ where $B = \{(x,y,z) | 1 \leq x \leq 2, \space 0 \leq y \leq 2, \space 1 \leq z \leq 3\}$

$$\iiint_B (x \space cos \space y + z) \space dV,$$ where $B = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq \pi, \space -1 \leq z \leq 1\}$

[Hide solution]

$0$

$$\iiint_B (z \space sin \space x + y^2) \space dV,$$ where $B = \{(x,y,z) | 0 \leq x \leq \pi, \space 0 \leq y \leq 1, \space -1 \leq z \leq 2\}$

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

$$\int_0^1 \int_1^2 \int_2^3 (x^2 + ln \space y + z) \space dx \space dy \space dz$$

[Hide Solution]

$$\int_0^1 \int_1^2 \int_2^3 (x^2 + ln \space y + z) \space dx \space dy \space dz = \frac{35}{6} + 2 \space ln 2$$

$$\int_0^1 \int_{-1}^1 \int_0^3 (ze^x + 2y) \space dx \space dy \space dz$$

$$\int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \space dx \space dy \space dz$$

[Hide solution]

$$\int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \space dx \space dy \space dz = 64 + 12 \space ln \space 3$$

$$\int_1^2 \int_{-2}^{-1} \int_0^1 \frac{x + y}{z} \space dx \space dy \space dz$$

Let $F$, $G$, and $H$ be continuous functions on $[a,b]$, $[c,d]$, and $[e,f]$, respectively, where $a, \space b, \space c, \space d, \space e$, and $f$ are real numbers such that $a < b, \space c < d$, and $e < f$. Show that

$$\int_a^b \int_c^d \int_e^f F (x) \space G (y) \space H(z) \space dz \space dy \space dx = \left(\int_a^b F(x) \space dx \right) \left(\int_c^d G(y) \space dy \right) \left(\int_e^f H(z) \space dz \right).$$

Let $F$, $G$, and $H$ be differential functions on $[a,b]$, $[c,d]$, and $[e,f]$, respectively, where $a, \space b, \space c, \space d, \space e$, and $f$ are real numbers such that $a < b, \space c < d$, and $e < f$. Show that

$$\int_a^b \int_c^d \int_e^f F' (x) \space G' (y) \space H'(z) \space dz \space dy \space dx = [F (b) - F (a)] \space [G(d) - G(c)] \space H(f) - H(e)].$$

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

$E = \{(x,y,z) | a \leq x \leq b, \space h_1 (x) \leq y \leq h_2 (x), \space e \leq z \leq f \}.$

$$\iiint_E (2x + 5y + 7z) \space dV,$$ where $E = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq -x + 1, \space 1 \leq z \leq 2\}$

[Hide solution]

$\frac{77}{12}$

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

$$\iiint_E (sin \space x + sin \space y) dV,$$ where $E = \{(x,y,z) | 0 \leq x \leq \frac{\pi}{2}, \space -cos \space x \leq y cos \space x, \space -1 \leq z \leq 1 \}$

[Hide Solution]

$2$

$$\iiint_E (xy + yz + xz ) dV$$ where $E = \{(x,y,z) | 0 \leq x \leq 1, \space -x^2 \leq y \leq x^2, \space 0 \leq z \leq 1 \}$

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

$$\iiint_E (x + 2yz) \space dV,$$ where $E = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq x, \space 0 \leq z \leq 5 - x - y \}$

[Hide Solution]

$\frac{430}{120}$

$$\iiint_E (x^3 + y^3 + z^3) \space dV,$$ where $E = \{(x,y,z) | 0 \leq x \leq 2, \space 0 \leq y \leq 2x, \space 0 \leq z \leq 4 - x - y \}$

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

[Hide Solution]

$0$

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

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

$E = \{(x,y,z) | g_1 (y) \leq x \leq g_2(y), \space c \leq y \leq d, \space e \leq z \leq f \}$.

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

[Hide Solution]

$-\frac{64}{105}$

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

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

[Hide Solution]

$\frac{11}{26}$

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

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

$E = \{(x,y,z) | g_1(y) \leq x \leq g_2(y), \space c \leq y \leq d, \space u_1(x,y) \leq z \leq u_2 (x,y) \}$

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

[Hide Solution]

$\frac{113}{450}$

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

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

[Hide Solution]

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

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

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

$E = \{(x,y,z) | (x,y) \in D, \space u_1 (x,y) x \leq z \leq u_2 (x,y) \}$, where $D$ is the projection of $E$ onto the $xy$-plane

$$\iint_D \left(\int_1^2 (x + y) \space dz \right) \space dA,$$ where $D = \{(x,y) | x^2 + y^2 \leq 1\}$

[Hide Solution]

$\frac{3\pi}{2}$

$$\iint_D \left(\int_1^3 x (z + 1)\space dz \right) \space dA,$$ where $D = \{(x,y) | x^2 -y^2 \geq 1, \space x \leq \sqrt{5}\}$

$$\iint_D \left(\int_0^{10-x-y} (x + 2z) \space dz \right) \space dA,$$ where $D = \{(x,y) | y \geq 0, \space x \geq 0, \space x + y \leq 10\}$

[Hide Solution]

$1250$

$$\iint_D \left(\int_0^{4x^2+4y^2} y \space dz \right) \space dA,$$ where $D = \{(x,y) | x^2 + y^2 \leq 4, \space y \geq 1, \space x \geq 0\}$

The solid $E$ bounded by $y^2 + z^2 = 9, \space z = 0$, and $x = 5$ is shown in the following figure. Evaluate the integral $$\iiint_E z \space dV$$ by integrating first with respect to $z$, then $y$, and then $x$.

[Hide Solution]

$$\int_0^5 \int_{-3}^3 \int_0^{\sqrt{9-y^2}} z \space dz \space dy \space dx = 90$$

The solid $E$ bounded by $y = \sqrt{x}, \space x = 4, \space y = 0$, and $z = 1$ is given in the following figure. Evaluate the integral $$\iiint_E xyz \space dV$$ by integrating first with respect to $x$, then $y$, and then $z$.

[T] The volume of a solid $E$ is given by the integral $$\int_{-2}^0 \int_x^0 \int_0^{x^2+y^2} dz \space dy \space dx.$$ Use a computer algebra system (CAS) to graph $E$ and find its volume. Round your answer to two decimal places.

[Hide Solution]

$V = 5.33$

[T] The volume of a solid $E$ is given by the integral $$\int_{-1}^0 \int_{-x^3}^0 \int_0^{1+\sqrt{x^2+y^2}} dz \space dy \space dx.$$ 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 $x, \space y,$ and $z$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x, \space y$, and $z$ is the arrangement of the numbers in one of the following orders: $y, \space z,$ and $x$ or $z, \space x,$ and $y$.

$$\int_0^1 \int_1^3 \int_2^4 (x^2z^2 + 1) dx \space dy \space dz$$

[Hide Solution]

$$\int_0^1 \int_1^3 \int_2^4 (y^2z^2 + 1) dz \space dx \space dy;$$ $$\int_0^1 \int_1^3 \int_2^4 (x^2z^2 + 1) dx \space dy \space dz$$

$$\int_0^3 \int_0^1 \int_0^{-x+1} (2x + 5y + 7z) dy \space dx \space dz$$

$$\int_0^1 \int_{-y}^y \int_0^{1-x^4-y^4} ln \space x dz \space dx \space dy$$

$$\int_{-1}^1 \int_0^1 \int_{-y^6}^{\sqrt{y}} (x + yz) dx \space dy \space dz$$

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 $a > 0$.

[Hide Solution]

$$V = \int_{-a}^a \int_{-\sqrt{a^2-z^2}}^{\sqrt{a^2-z^2}} \int_{\sqrt{x^2+z^2}}^{a^2} dy \space dx \space dz$$

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 $x + 0, \space x = 1, \space y = 0, \space y = 3, \space z = 0$, and $z = 5$.

[Hide Solution]

$\frac{9}{2}$

Find the average value of the function $f(x,y,z) = xyz$ 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 $xy$-plane is bounded by $x = sin^{-1} y, \space y = 0$, and $x = \frac{\pi}{2}$.

Consider the pyramid with the base in the $xy$-plane of $[-2,2] \times [-2,2]$ and the vertex at the point $(0,0,8)$.

a. Show that the equations of the planes of the lateral faces of the pyramid are $4y + z = 8, \space 4y - z = -8, \space 4x + z = 8$, and $-4x + z = 8$.

b. Find the volume of the pyramid.

[Hide Solution]

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

Consider the pyramid with the base in the $xy$-plane of $[-3,3] \times [-3,3]$ and the vertex at the point $(0,0,9)$.

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

b. 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.

a. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z$, then with $y$, and then with $x$.

b. Rewrite the integral in part a. as an equivalent integral in five other orders.

[Hide Solution]

$$a. \space \int_0^4 \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dz \space dy \space dx; \space b. \space \int_0^2 \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dz \space dx \space dy,$$

$$\int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dy \space dx \space dz, \space \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dy \space dz \space dx,$$

$$\int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dx \space dy \space dz, \space \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dx \space dz \space dy,$$

The solid $E$ bounded by the sphere of equation $9x^2 + 4y^2 + z^2 = 1$ and located in the first octant is represented in the following figure.

a. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z$ then with $y$ and then with $x$.

b. Rewrite the integral in part a. as an equivalent integral in five other orders.

Find the volume of the prism with vertices $(0,0,0), \space (2,0,0), \space (2,3,0), \space (0,3,0), \space (0,0,1)$, and $(2,0,1)$.

[Hide Solution]

$3$

Find the volume of the prism with vertices $(0,0,0), \space (4,0,0), \space (4,6,0), \space (0,6,0), \space (0,0,1)$, and $(4,0,1)$.

The solid $E$ bounded by $z = 10 - 2x - y$ and situated in the first octant is given in the following figure. Find the volume of the solid.

[Hide Solution]

$\frac{250}{3}$

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.

The midpoint rule for the triple integral $$\iiint_B f(x,y,z) dV$$ 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 $$\sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(\bar{x_i}, \bar{y_j}, \bar{z_k}) \Delta V,$$ where $(\bar{x_i}, \bar{y_j}, \bar{z_k})$ is the center of the box $B_{ijk}$ and $\Delta V$ is the volume of each subbox. Apply the midpoint rule to approximate $$\iiint_B x^2 dV$$ over the solid $B = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq 1, \space 0 \leq z \leq 1 \}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

[Hide Solution]

$\frac{5}{16} \approx 0.313$

[T]

a. Apply the midpoint rule to approximate $$\iiint_B e^{-x^2} dV$$ over the solid $B = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq 1, \space 0 \leq z \leq 1 \}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

b. 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 $T (x,y,z) = xz + 5z + 10$. Find the average temperature over the solid.

[Hide Solution]

$\frac{35}{2}$

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 $T(x,y,z) = x + y + xy$. Find the average temperature over the solid.

Show that the volume of a right square pyramid of height $h$ and side length $a$ is $v = \frac{ha^2}{3}$ by using triple integrals.

Show that the volume of a regular right hexagonal prism of edge length $a$ is $\frac{3a^3 \sqrt{3}}{2}$ 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 $\rho (x,y,z)$, then the total charge inside the solid is defined as the triple integral $$\iiint_E \rho (x,y,z) dV.$$ 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 $E$.

Contributors

• OpenStax Calculus 3
• Apex Calculus
• Edited by Paul Seeburger (Monroe Community College)