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Mathematics LibreTexts

7.5E: Excersies

  • Page ID
    25944
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    Exercise \(\PageIndex{1}\)

    In the following exercises, evaluate the triple integrals \[\iiint_E f(x,y,z) dV\] over the solid \(E\).

    1. \(f(x,y,z) = z, \space B = \{(x,y,z) | x^2 + y^2 \leq 9, \space x \leq 0, \space y \leq 0, \space 0 \leq z \leq 1\}\)

    A quarter section of a cylinder with height 1 and radius 3.

    Answer:

    \(\frac{9\pi}{8}\)

    2. \(f(x,y,z) = xz^2, \space B = \{(x,y,z) | x^2 + y^2 \leq 16, \space x \geq 0, \space y \leq 0, \space -1 \leq z \leq 1\}\)

    3. \(f(x,y,z) = xy, \space B = \{(x,y,z) | x^2 + y^2 \leq 1, \space x \geq 0, \space x \geq y, \space -1 \leq z \leq 1\}\)

    A wedge with radius 1, height 1, and angle pi/4.

    Answer:

    \(\frac{1}{8}\)st.

    4. \(f(x,y,z) = x^2 + y^2, \space B = \{(x,y,z) | x^2 + y^2 \leq 4, \space x \geq 0, \space x \leq y, \space 0 \leq z \leq 3\}\)

    5. \(f(x,y,z) = e^{\sqrt{x^2+y^2}}, \space B = \{(x,y,z) | 1 \leq x^2 + y^2 \leq 4, \space y \leq 0, \space x \leq y\sqrt{3}, \space 2 \leq z \leq 3 \}\)

    Answer:

    \(\frac{\pi e^2}{6}\)

    6. \(f(x,y,z) = \sqrt{x^2 + y^2}, \space B = \{(x,y,z) | 1 \leq x^2 + y^2 \leq 9, \space y \leq 0, \space 0 \leq z \leq 1\}\)

    Exercise \(\PageIndex{2}\)

    1.

    a. Let \(B\) be a cylindrical shell with inner radius \(a\) outer radius \(b\), and height \(c\) where \(0 < a < b\) and \(c>0\). Assume that a function \(F\) defined on \(B\) can be expressed in cylindrical coordinates as \(F(x,y,z) = f(r) + h(z)\), where \(f\) and \(h\) are differentiable functions. If \[\int_a^b \bar{f} (r) dr = 0\] and \(\bar{h}(0) = 0\), where \(\bar{f}\) and \(\bar{h}\) are antiderivatives of \(f\) and \(h\), respectively, show that

    \[\iiint_B F(x,y,z) dV = 2\pi c (b\bar{f} (b) - a \bar{f}(a)) + \pi(b^2 - a^2) \bar{h} (c).\]

    b. Use the previous result to show that

    \[\iiint_B \left(z + sin \sqrt{x^2 + y^2}\right) dx \space dy \space dz = 6 \pi^2 ( \pi - 2),\]

    where \(B\) is a cylindrical shell with inner radius \(\pi\) outer radius \(2\pi\), and height \(2\).

    2.

    a. Let \(B\) be a cylindrical shell with inner radius \(a\) outer radius \(b\) and height \(c\) where \(0 < a < b\) and \(c > 0\). Assume that a function \(F\) defined on \(B\) can be expressed in cylindrical coordinates as F(x,y,z) = f(r) g(\theta) f(z)\), where \(f, \space g,\) and \(h\) are differentiable functions. If \[\int_a^b \tilde{f} (r) dr = 0,\] where \(\tilde{f}\) is an antiderivative of \(f\), show that

    \[\iiint_B F (x,y,z)dV = [b\tilde{f}(b) - a\tilde{f}(a)] [\tilde{g}(2\pi) - \tilde{g}(0)] [\tilde{h}(c) - \tilde{h}(0)],\]

    where \(\tilde{g}\) and \(\tilde{h}\) are antiderivatives of \(g\) and \(h\), respectively.

    b. Use the previous result to show that \[\iiint_B z \space sin \sqrt{x^2 + y^2} dx \space dy \space dz = - 12 \pi^2,\] where \(B\) is a cylindrical shell with inner radius \(\pi\) outer radius \(2\pi\), and height \(2\).

    Exercise \(\PageIndex{3}\)

    In the following exercises, the boundaries of the solid \(E\) are given in cylindrical coordinates.

    a. Express the region \(E\) in cylindrical coordinates.

    b. Convert the integral \[\iiint_E f(x,y,z) dV\] to cylindrical coordinates.

    1. E is bounded by the right circular cylinder \(r = 4 \space sin \space \theta\), the \(r\theta\)-plane, and the sphere \(r^2 + z^2 = 16\).

    Answer:

    a. \(E = \{(r,\theta,z) | 0 \leq \theta \leq \pi, \space 0 \leq r \leq 4 \space sin \space \theta, \space 0 \leq z \leq \sqrt{16 - r^2}\}\)

    b. \[\int_0^{\pi} \int_0^{4 \space sin \space \theta} \int_0^{\sqrt{16-r^2}} f(r,\theta, z) r \space dz \space dr \space d\theta\]

    2. \(E\) is bounded by the right circular cylinder \(r = cos \space \theta\), the \(r\theta\)-plane, and the sphere \(r^2 + z^2 = 9\).

    3. \(E\) is located in the first octant and is bounded by the circular paraboloid \(z = 9 - 3r^2\), the cylinder \(r = \sqrt{r}\), and the plane \(r(cos \space \theta + sin \space \theta) = 20 - z\).

    Answer:

    a. \(E = \{(r,\theta,z) |0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq r \leq \sqrt{3}, \space 9 - r^2 \leq z \leq 10 - r(cos \space \theta + sin \space \theta)\};

    b. \[\int_0^{\pi/2} \int_0^{\sqrt{3}} \int_{9-r^2}^{10-r(cos \space \theta + sin \space \theta)} f(r,\theta,z) r \space dz \space dr \space d\theta\]

    4. \(E\) is located in the first octant outside the circular paraboloid \(z = 10 - 2r^2\) and inside the cylinder \(r = \sqrt{5}\) and is bounded also by the planes \(z = 20\) and \(\theta = \frac{\pi}{4}\).

    Exercise \(\PageIndex{4}\)

    In the following exercises, the function \(f\) and region \(E\) are given.

    a. Express the region \(E\) and the function \(f\) in cylindrical coordinates.

    b. Convert the integral \[\iiint_B f(x,y,z) dV\] into cylindrical coordinates and evaluate it.

    1. \(f(x,y,z) = x^2 + y^2\), \(E = \{(x,y,z) | 0 \leq x^2 + y^2 \leq 9, \space x \geq 0, \space y \geq 0, \space 0 \leq z \leq x + 3\}\)

    Answer:

    a. \(E = \{(r,\theta,z) | 0 \leq r \leq 3, \space 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq z \leq r \space cos \space \theta + 3\},\) f(r,\theta,z) = \frac{1}{r \space cos \space \theta + 3};

    b. \[\int_0^3 \int_0^{\pi/2} \int_0^{r \space cos \space \theta+3} \frac{r}{r \space cos \space \theta + 3} dz \space d\theta \space dr = \frac{9\pi}{4}\]

    2. \(f(x,y,z) = x^2 + y^2, \space E = \{(x,y,z) |0 \leq x^2 + y^2 \leq 4, \space y \geq 0, \space 0 \leq z \leq 3 - x\)

    \(f(x,y,z) = x, \space E = \{(x,y,z) | 1 \leq y^2 + z^2 \leq 9, \space 0 \leq x \leq 1 - y^2 - z^2\}\)

    Answer:

    a. \(y = r \space cos \space \theta, \space z = r \space sin \space \theta, \space x = z,\space E = \{(r,\theta,z) | 1 \leq r \leq 3, \space 0 \leq \theta \leq 2\pi, \space 0 \leq z \leq 1 - r^2\}, \space f(r,\theta,z) = z\);

    b. \[\int_1^3 \int_0^{2\pi} \int_0^{1-r^2} z r \space dz \space d\theta \space dr = \frac{356 \pi}{3}\]

    3. \(f(x,y,z) = y, \space E = \{(x,y,z) | 1 \leq x^2 + z^2 \leq 9, \space 0 \leq y \leq 1 - x^2 - z^2 \}\)

    Exercise \(\PageIndex{5}\)

    In the following exercises, find the volume of the solid \(E\) whose boundaries are given in rectangular coordinates.

    1. \(E\) is above the \(xy\)-plane, inside the cylinder \(x^2 + y^2 = 1\), and below the plane \(z = 1\).

    Answer:

    \(\pi\)

    2. \(E\) is below the plane \(z = 1\) and inside the paraboloid \(z = x^2 + y^2\).

    3. \(E\) is bounded by the circular cone \(z = \sqrt{x^2 + y^2}\) and \(z = 1\).

    Answer:

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

    4. \(E\) is located above the \(xy\)-plane, below \(z = 1\), outside the one-sheeted hyperboloid \(x^2 + y^2 - z^2 = 1\), and inside the cylinder \(x^2 + y^2 = 2\).

    5. \(E\) is located inside the cylinder \(x^2 + y^2 = 1\) and between the circular paraboloids \(z = 1 - x^2 - y^2\) and \(z = x^2 + y^2\).

    Answer:

    \(\pi\)

    6. \(E\) is located inside the sphere \(x^2 + y^2 + z^2 = 1\), above the \(xy\)-plane, and inside the circular cone \(z = \sqrt{x^2 + y^2}\).

    7. \(E\) is located outside the circular cone \(x^2 + y^2 = (z - 1)^2\) and between the planes \(z = 0\) and \(z = 2\).

    Answer:

    \(\frac{4\pi}{3}\)

    8. \(E\) is located outside the circular cone \(z = 1 - \sqrt{x^2 + y^2}\), above the \(xy\)-plane, below the circular paraboloid, and between the planes \(z = 0\) and \(z = 2\).

    Exercise \(\PageIndex{6}\)

    1. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates \[\int_{-\pi/2}^{\pi/2} \int_0^1 \int_{r^2}^r r \space dz \space dr \space d\theta.\] Find the volume \(V\) of the solid. Round your answer to four decimal places.

    Answer:

    \(V = \frac{pi}{12} \approx 0.2618\)

    A quarter section of an ellipsoid with width 2, height 1, and depth 1.

    2. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates \[\int_0^{\pi/2} \int_0^1 \int_{r^4}^r r \space dz \space dr \space d\theta.\] Find the volume \(E\) of the solid Round your answer to four decimal places.

    Exercise \(\PageIndex{7}\)

    1. Convert the integral \[\int_0^1 \int_{-\sqrt{1-z^2}}^{\sqrt{1-y^2}} \int_{x^2+y^2}^{\sqrt{x^2+y^2}} xz \space dz \space dx \space dy\] into an integral in cylindrical coordinates.

    Answer:

    \[\int_0^1 \int_0^{\pi} \int_{r^2}^r zr^2 \space cos \space \theta \space dz \space d\theta \space dr\

    2. Convert the integral \[\int_0^2 \int_0^x \int_0^1 (xy + z) dz \space dx \space dy\] into an integral in cylindrical coordinates.

    Exercise \(\PageIndex{8}\)

    In the following exercises, evaluate the triple integral \[\iiint_B f(x,y,z)dV\] over the solid \(B\).

    1. \(f(x,y,z) = 1, \space B = \{(x,y,z) | x^2 + y^2 + z^2 \leq 90, \space z \geq 0\}\)

    A filled-in half-sphere with radius 3 times the square root of 10.

    Answer:

    \(180 \pi \sqrt{10}\)

    2. \(f(x,y,z) = 1 - \sqrt{x^2 + y^2 + z^2}, \space B = \{(x,y,z) | x^2 + y^2 + z^2 \leq 9, \space y \geq 0, \space z \geq 0\}\)

    A quarter section of an ovoid with height 8, width 8 and length 18.

    3. \(f(x,y,z) = \sqrt{x^2 + y^2}, \space B \) is bounded above by the half-sphere \(x^2 + y^2 + z^2 = 9\) with \(z \geq 0\) and below by the cone \(2z^2 = x^2 + y^2\).

    Answer:

    \(\frac{81\pi(\pi - 2)}{16}\)

    4. \(f(x,y,z) = \sqrt{x^2 + y^2}, \space B \) is bounded above by the half-sphere \(x^2 + y^2 + z^2 = 16\) with \(z \geq 0\) and below by the cone \(2z^2 = x^2 + y^2\).

    Exercise \(\PageIndex{9}\)

    Show that if \(F ( \rho,\theta,\varphi) = f(\rho)g(\theta)h(\varphi)\) is a continuous function on the spherical box \(B = \{(\rho,\theta,\varphi) | a \leq \rho \leq b, \space \alpha \leq \theta \leq \beta, \space \gamma \leq \varphi \leq \psi\}\), then

    \[\iiint_B F \space dV = \left(\int_a^b \rho^2 f(\rho) \space dr \right) \left( \int_{\alpha}^{\beta} g (\theta) \space d\theta \right)\left( \int_{\gamma}^{\psi} h (\varphi) \space sin \space \varphi \space d\varphi \right).\]

    Exercise \(\PageIndex{10}\)

    1.

    a. A function \(F\) is said to have spherical symmetry if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as \(F(x,y,z) = f(\rho)\), where \(\rho = \sqrt{x^2 + y^2 + z^2}\). Show that \[\iiint_B F(x,y,z) dV = 2\pi \int_a^b \rho^2 f(\rho) d\rho,\] where \(B\) is the region between the upper concentric hemispheres of radii \(a\) and \(b\) centered at the origin, with \(0 < a < b\) and \(F\) a spherical function defined on \(B\).

    b. Use the previous result to show that \[\iiint_B (x^2 + y^2 + z^2) \sqrt{x^2 + y^2 + z^2} dV = 21 \pi,\] where \(B = \{(x,y,z) | 1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0\}\).

    2.

    a. Let \(B\) be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where \(0 < a < b\). Consider F a function defined on B whose form in spherical coordinates \((\rho,\theta,\varphi)\) is \(F(x,y,z) = f(\rho)cos \space \varphi\). Show that if \(g(a) = g(b) = 0\) and \[\int_a^b h (\rho)d\rho = 0,\] then \[\iiint_B F(x,y,z)dV = \frac{\pi^2}{4} [ah(a) - bh(b)],\] where \(g\) is an antiderivative of \(f\) and \(h\) is an antiderivative of \(g\).

    b. Use the previous result to show that \[\iiint_B = \frac{z \space cos \sqrt{x^2 + y^2 + z^2}}{\sqrt{x^2 + y^2 + z^2}}dV = \frac{3\pi^2}{2},\] where \(B\) is the region between the upper concentric hemispheres of radii \(\pi\) and \(2\pi\) centered at the origin and situated in the first octant.

    Exercise \(\PageIndex{11}\)

    In the following exercises, the function \(f\) and region \(E\) are given.

    a. Express the region \(E\) and function \(f\) in cylindrical coordinates.

    b. Convert the integral \[\iiint_B f(x,y,z)dV\] into cylindrical coordinates and evaluate it.

    1. \(f(x,y,z) = z; \space E = \{(x,y,z) | 0 \leq x^2 + y^2 + z^2 \leq 1, \space z \geq 0\}\)

    2. \(f(x,y,z) = x + y; \space E = \{(x,y,z) | 1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0, \space y \geq 0\}\)

    Answer:

    a. \(f(\rho,\theta, \varphi) = \rho \space sin \space \varphi \space (cos \space \theta + sin \space \theta), \space E = \{(\rho,\theta,\varphi) | 1 \leq \rho \leq 2, \space 0 \leq \theta \leq \pi, \space 0 \leq \varphi \leq \frac{\pi}{2}\}\);

    b. \[\int_0^{\pi} \int_0^{\pi/2} \int_1^2 \rho^3 cos \space \varphi \space sin \space \varphi \space d\rho \space d\varphi \space d\theta = \frac{15\pi}{8}\]

    3. \(f(x,y,z) = 2xy; \space E = \{(x,y,z) | \sqrt{x^2 + y^2} \leq z \leq \sqrt{1 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\}\)

    4. \(f(x,y,z) = z; \space E = \{(x,y,z) | x^2 + y^2 + z^2 - 2x \leq 0, \space \sqrt{x^2 + y^2} \leq z\}\)

    Answer:

    a. \(f(\rho,\theta,\varphi) = \rho \space cos \space \varphi; \space E = \{(\rho,\theta,\varphi) | 0 \leq \rho \leq 2 \space cos \space \varphi, \space 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq \varphi \leq \frac{\pi}{4}\}\);

    b. \[\int_0^{\pi/2} \int_0^{\pi/4} \int_0^{2 \space cos \space \varphi} \rho^3 sin \space \varphi \space cos \space \varphi \space d\rho \space d\varphi \space d\theta = \frac{7\pi}{24}\]

    Exercise \(\PageIndex{12}\)

    In the following exercises, find the volume of the solid \(E\) whose boundaries are given in rectangular coordinates.

    1. \(E = \{ (x,y,z) | \sqrt{x^2 + y^2} \leq z \leq \sqrt{16 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\}\)

    2. \(E = \{ (x,y,z) | x^2 + y^2 + z^2 - 2z \leq 0, \space \sqrt{x^2 + y^2} \leq z\}\)

    Answer:

    \(\frac{\pi}{4}\)

    3. Use spherical coordinates to find the volume of the solid situated outside the sphere \(\rho = 1\) and inside the sphere \(\rho = cos \space \varphi\), with \(\varphi \in [0,\frac{\pi}{2}]\).

    4. Use spherical coordinates to find the volume of the ball \(\rho \leq 3\) that is situated between the cones \(\varphi = \frac{\pi}{4}\) and \(\varphi = \frac{\pi}{3}\).

    Answer:

    \(9\pi (\sqrt{2} - 1)\)

    5. Convert the integral \[\int_{-4}64 \int_{-\sqrt{16-y^2}}^{\sqrt{16-y^2}} \int_{-\sqrt{16-x^2-y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^2 dz \space dy \space dx\] into an integral in spherical coordinates.

    Answer:

    \[\int_0^{\pi/2} \int_0^{\pi/2} \int_0^4 \rho^6 sin \space \varphi \space d\theta\]

    6. Convert the integral \[\int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{\sqrt{16-x^2-y^2}} dz \space dy \space dx\] into an integral in spherical coordinates and evaluate it.

    Exercise \(\PageIndex{13}\)

    1. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates \[\int_{\pi/2}^{\pi} \int_{5\pi}^{\pi/6} \int_0^2 \rho^2 sin \space \varphi \space d\rho \space d\varphi \space d\theta.\] Find the volume \(V\) of the solid. Round your answer to three decimal places.

    Answer:

    \(V = \frac{4\pi\sqrt{3}}{3} \approx 7.255\)

    A sphere of radius 1 with a hole drilled into it of radius 0.5.

    2. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as \[\int_0^{2\pi} \int_{3\pi/4}^{\pi/4} \int_0^1 \rho^2 sin \space \varphi \space d\rho \space d\varphi \space d\theta.\] Find the volume \(V\) of the solid. Round your answer to three decimal places.

    3. [T] Use a CAS to evaluate the integral \[ \iiint_E (x^2 + y^2) dV\] where \(E\) lies above the paraboloid \(z = x^2 + y^2\) and below the plane \(z = 3y\).

    Answer:

    \(\frac{343\pi}{32}\)

    4. [T]

    a. Evaluate the integral \[\iiint_E e^{\sqrt{x^2+y^2+z^2}}dV,\] where \(E\) is bounded by spheres \(4x^2 + 4y^2 + 4z^2 = 1\) and \(x^2 + y^2 + z^2 = 1\).

    b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.

    Exercise \(\PageIndex{14}\)

    Express the volume of the solid inside the sphere \(x^2 + y^2 + z^2 = 16\) and outside the cylinder \(x^2 + y^2 = 4\)as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

    Exercise \(\PageIndex{15}\)

    1. The power emitted by an antenna has a power density per unit volume given in spherical coordinates by \(p(\rho,\theta,\varphi) = \frac{P_0}{\rho^2} cos^2 \theta \space sin^4 \varphi\), where \(P_0\) is a constant with units in watts. The total power within a sphere \(B\) of radius \(r\) meters is defined as \[P = \iiint_B p(\rho,\theta,\varphi) \space dV.\] Find the total power \(P\).

    Answer:

    \(P = \frac{32P_0 \pi}{3}\) watts

    2. Use the preceding exercise to find the total power within a sphere \(B\) of radius 5 meters when the power density per unit volume is given by \(p(\rho, \theta,\varphi) = \frac{30}{\rho^2} cos^2 \theta \space sin^4 \varphi\).

    3. A charge cloud contained in a sphere \(B\) of radius r centimeters centered at the origin has its charge density given by \(q(x,y,z) = k\sqrt{x^2 + y^2 + z^2}\frac{\mu C}{cm^3}\), where \(k > 0\).

    The total charge contained in \(B\) is given by \[Q = \iiint_B q(x,y,z) dV.\] Find the total charge \(Q\).

    Answer:

    \(Q = kr^4 \pi \mu C\)

    4. Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is \(q(x,y,z) = 20 \sqrt{x^2 + y^2 + z^2} \frac{\mu C}{cm^3}\).