3.5E: Triple Integrals (Exercises)

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 $\displaystyle \iiint_Q \,dV$ means.

Answer:

$\displaystyle \iiint_Q \,dV$ = 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 $(1,1)$.

Answer:

V = $\displaystyle \int_{-1}^{1}\int_{-1}^{1}\int_{2x+y}^{8-x^2-y^2} \,dz\,dy\,dx\quad$ $=\quad\dfrac{88}{3}\,\text{units}^3$

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 $(\pi,\pi)$.

Answer:

V = $\displaystyle \int_{0}^{\pi}\int_{0}^{x}\int_{\sin x\cos y}^{\cos x\sin y + 2} \,dz\,dy\,dx\quad$ $=\quad\left(\pi^2 - \pi\right)\,\text{units}^3\quad$ $\approx 6.72801\,\text{units}^3$

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(s) 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 $dz\,dy\,dx$.

Answer:

V = $\displaystyle \int_{0}^{3}\int_{0}^{1-\frac{x}{3}}\int_{0}^{2 - \frac{2}{3}x-2y} \,dz\,dy\,dx\quad$ $=\quad 1\,\text{unit}^3$

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 orders $dy\,dz\,dx$ and $dz\,dy\,dx$ to verify that you obtain the same volume either way.

Answer:

V = $\displaystyle \int_{0}^{2}\int_{0}^{2}\int_{-\sqrt{2z}}^{-z} \,dy\,dz\,dx\quad$ $=\quad \dfrac{4}{3}\,\text{unit}^3$
V = $\displaystyle \int_{0}^{2}\int_{-2}^{0}\int_{\frac{y^2}{2}}^{-y} \,dz\,dy\,dx\quad$ $=\quad \dfrac{4}{3}\,\text{unit}^3$

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 in the orders:  $dz\,dy\,dx$, $dx\,dy\,dz$, and  $dy\,dz\,dx$ .

13. $D$ is bounded by the planes $x=2,y=1,z=0$ and $z=2x+4y-4$.
Evaluate the triple integral with orders $dz\,dy\,dx$ and $dx\,dy\,dz$ to verify that you obtain the same volume either way.

Answer:

V = $\displaystyle \int_{0}^{2}\int_{1-\frac{x}{2}}^{1}\int_{0}^{2x+4y-4} \,dz\,dy\,dx\quad$ $=\quad\dfrac{4}{3}\,\text{units}^3$
V = $\displaystyle \int_{0}^{4}\int_{\frac{z}{4}}^{1}\int_{(z-4y+4)/2}^{2} \,dx\,dy\,dz\quad$ $=\quad\dfrac{4}{3}\,\text{units}^3$

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.

Answer:

V = $\displaystyle \int_{0}^{1}\int_{0}^{1-x^2}\int_{0}^{\sqrt{1-y}} \,dz\,dy\,dx\quad$
V = $\displaystyle \int_{0}^{1}\int_{0}^{x}\int_{0}^{1-x^2} \,dy\,dz\,dx + \displaystyle \int_{0}^{1}\int_{x}^{1}\int_{0}^{1-z^2} \,dy\,dz\,dx$
The first one is easier since it only requires evaluation of a single integral, although both can be evaluated fairly easily.

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

Evaluating General Triple Integrals

In exercises 17 - 20, evaluate the triple integrals over the rectangular solid box $B$.

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

Answer:

$192$

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

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

Answer:

$0$

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

In Exercises 21 - 24, evaluate the triple integral.

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

Answer:

$8$

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

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

Answer:

$\pi$

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

Average Value of a Function

25. 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 $z = 5$.

Answer:

$\frac{9}{2}$

26. 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.

Approximating Triple Integrals

27. The midpoint rule for the triple integral $\displaystyle \iiint_B f(x,y,z) \,dV$ 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 $$\sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(\bar{x_i}, \bar{y_j}, \bar{z_k}) \Delta V,\nonumber$$ 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 $$\iiint_B x^2 \,dV\nonumber$$ over the solid $B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 1, \space 0 \leq z \leq 1 \big\}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

Answer:

$\displaystyle \iiint_B f(x,y,z) \,dV\quad$ $\approx\quad\frac{5}{16} \approx 0.313$

28. [T] a. Apply the midpoint rule to approximate $$\iiint_B e^{-x^2} \,dV\nonumber$$ 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 $n = 3,4, ..., 10$.

Applications

29. Suppose that the temperature in degrees Celsius at a point $(x,y,z)$ of a solid $E$ bounded by the coordinate planes and the plane $x + y + z = 5$ is given by: $$T (x,y,z) = xz + 5z + 10\nonumber$$ Find the average temperature over the solid.

Answer:

$17.5^{\circ}$ C

30. Suppose that the temperature in degrees Fahrenheit at a point $(x,y,z)$ of a solid $E$ bounded by the coordinate planes and the plane $x + y + z = 5$ is given by: $$T(x,y,z) = x + y + xy\nonumber$$ Find the average temperature over the solid.

31. 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 $\displaystyle \iiint_E \rho (x,y,z) \,dV.$ 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 $E$.

Answer:

Total Charge inside the Solid $E \quad=\quad$ $\displaystyle \int_{-\sqrt{5}}^{\sqrt{5}}\int_{-\sqrt{5-y^2}}^{\sqrt{5-y^2}}\int_{y^2+z^2-5}^{5 - y^2 - z^2} \sqrt{x^2+y^2+z^2}\,dx\,dz\,dy$

32. Show that the volume of a regular right hexagonal pyramid of edge length $a$ is $\dfrac{a^3 \sqrt{3}}{2}$ by using triple integrals.

Contributors

• Problems 17 - 20 and 25 - 32 are from Section 15.4, OpenStax Calculus 3 by

  Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

• Problems 1 - 16 and 21 - 24 are from Apex Calculus, Section 13.6.
• Edited by Paul Seeburger (Monroe Community College)