9.4E:Exercises

Exercise \(\PageIndex{1}\)

In the following exercises, evaluate the triple integrals over the rectangular solid box \(B\).

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

Answer

\(192\)

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

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

Answer

\(0\)

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

Exercise \(\PageIndex{2}\)

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

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

Answer

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

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

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

Answer

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

4. \[\int_1^2 \int_{-2}^{-1} \int_0^1 \frac{x + y}{z} \space dx \space dy \space dz\]

Exercise \(\PageIndex{3}\)

1. 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

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

2. 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

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

Exercise \(\PageIndex{4}\)

In the following exercises, evaluate the triple integrals over the bounded region \(E = \{(x,y,z) | a \leq x \leq b, \space h_1 (x) \leq y \leq h_2 (x), \space e \leq z \leq f \}.\)

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

Answer

\(\frac{77}{12}\)

2. \[\iiint_E (y \space ln \space x + z) \space dV,\] 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 \(E = \{(x,y,z) | 0 \leq x \leq \frac{\pi}{2}, \space -cos \space x \leq y cos \space x, \space -1 \leq z \leq 1 \}\)

Answer

\(2\)

3. \[\iiint_E (xy + yz + xz ) dV\] 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 \(E\).

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

Answer

\(\frac{430}{120}\)

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

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

Answer

\(0\)

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

Exercise \(\PageIndex{5}\)

In the following exercises, evaluate the triple integrals over the bounded region \(E\) of the form \(E = \{(x,y,z) | g_1 (y) \leq x \leq g_2(y), \space c \leq y \leq d, \space e \leq z \leq f \}\).

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

Answer

\(-\frac{64}{105}\)

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

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

Answer

\(\frac{11}{26}\)

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

Exercise \(\PageIndex{6}\)

In the following exercises, evaluate the triple integrals over the bounded region \(E = \{(x,y,z) | g_1(y) \leq x \leq g_2(y), \space c \leq y \leq d, \space u_1(x,y) \leq z \leq u_2 (x,y) \}\)

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

Answer

\(\frac{113}{450}\)

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

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

Answer

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

4. \[\iiint_E (x + y) \space dV,\] 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

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

Answer

\(\frac{3\pi}{2}\)

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

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

Answer

\(1250\)

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

Exercise \(\PageIndex{7}\)

1. 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 \(x\).

Answer

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

2. 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 \(z\).

3. [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.

Answer

\(V = 5.33\)

4. [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.

Exercise \(\PageIndex{8}\)

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 \(y\).

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

Answer

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

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

3. \[\int_0^1 \int_{-y}^y \int_0^{1-x^4-y^4} ln \space x dz \space dx \space dy\]

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

Exercise \(\PageIndex{9}\)

1. 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 \(a > 0\).

Answer

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

2. Set up the integral that gives the volume of the solid \(E\) bounded by \(x = y^2 + z^2\) and \(x = a^2\), where \(a > 0\).

Exercise \(\PageIndex{10}\) Average value

1. 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

\(\frac{9}{2}\)

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.

Exercise \(\PageIndex{11}\)

1. 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 \(x = \frac{\pi}{2}\).

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.

Answer

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

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.

4. 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.

Answer

\[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,\]

5. 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.

6. 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 \((2,0,1)\).

Answer

\(3\)

7. 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 \((4,0,1)\).

8. 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.

Answer

\(\frac{250}{3}\)

9. 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.

Exercise \(\PageIndex{12}\)

1. 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.

Answer

\(\frac{5}{16} \approx 0.313\)

2. [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 \(n = 3,4, ..., 10\).

Exercise \(\PageIndex{13}\)

1. 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.

Answer

\(\frac{35}{2}\

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.

Exercise \(\PageIndex{14}\)

1. 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.
2. 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.
3. 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.

Exercise \(\PageIndex{15}\)

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 \(E\).