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Mathematics LibreTexts

14.4E: Triple Integrals (Exercises)

  • Page ID
    13806
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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 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 \((1,1)\).

    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 \(dz\,dy\,dx\).
    13609.PNG

    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 \(dx\,dy\,dz\).
    13610.PNG

    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.
    13611.PNG

    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.
    13612.PNG

    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.
    13613.PNG

    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 \(dz\,dy\,dx\).
    13614.PNG

    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. 
    13615.PNG

    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 \(dx\,dy\,dz\).
    13616.PNG

     

    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)