# 4.11: Triple Integrals (Exercises)

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## 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
to**surface**, then from__surface__to**curve**and then from__curve__to__point__.**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 \((1,1)\).

**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.

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)