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Mathematics LibreTexts

7.4E:Exercises

  • Page ID
    25943
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    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