14.4E: 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)