?A surface \(S\) consists of the cylinder \(x^{2}+y^{2}=1,-1 \leqslant z \leqslant 1\)

Let S be the surface of the box enclosed by the planes \(x=\pm 1\), \(y=\pm 1, \ z=\pm 1\). Approximate \(\iint_{\mathrm{s}} \cos (x+2 y+3 z) \ d S\) by using a Riemann sum as in Definition 1, taking the patches \(S_{i j}\) to be the squares that are the faces of the box S and the points \(P_{i j}^{*}\) to be the centers of the squares.

A surface \(S\) consists of the cylinder \(x^{2}+y^{2}=1,-1 \leqslant z \leqslant 1\), together with its top and bottom disks. Suppose you know that \(f\) is a continuous function with \(f(\pm 1,0,0)=2 \quad f(0, \pm 1,0)=3 \quad f(0,0, \pm 1)=4\) Estimate the value of \(\iint_{s} f(x, y, z) d S\) by using a Riemann sum, taking the patches \(S_{i j}\) to be four quarter-cylinders and the top and bottom disks. Equation Transcription: Text Transcription: S x^2+y^2=1, -1 leqslant z leqslant 1 f f(pm 1,0,0)=2 f(0,pm 1,0)=3 f(0,0,pm 1)=4 ?_s f(x,y,z)dS S_ij

Let \(H\) be the hemisphere \(x^{2}+y^{2}+z^{2}=50, z \geqslant 0\), and suppose \(f\) is a continuous function with \(f(3,4,5)=7, f(3,-4,5)=8, f(-3,4,5)=9\), and \(f(-3,-4,5)=12\). By dividing \(H\) into four patches, estimate the value of \(\iint_{H} f(x, y, z) d S\) Equation Transcription: Text Transcription: H x^2+y^2+z^2=50, x geqslant0 f f(3,4,5)=7, f(3,-4,5)=8, f(-3,4,5)=9 and f(-3,-4, 5)=12 ?_H f(x,y,z)dS

Suppose that \(f(x, y, z)=g\left(\sqrt{x^{2}+y^{2}+z^{2}}\right)\), where g is a function of one variable such that \(g(2)=-5\). Evaluate \(\iint_{s} f(x, y, z) d S\), where S is the sphere \(x^{2}+y^{2}+z^{2}=4\). Equation Transcription: ? Text Transcription: f(x,y,z) = g(sqrt x^2 + y^2 + z^2) g(2) = -5 Double Integral_s f(x,y,z) dS x^2 + y^2 + z^2 = 4

Evaluate the surface integral. \(\iint_{S}(x+y+z) d S\), S is the parallelogram with parametric equations \(x=u+v\), \(y=u-v, z=1+2 u+v, 0 \leqslant \mathrm{u} \leqslant 2,0 \leqslant \mathrm{v} \leqslant 1\) Equation Transcription: ? 0 ? u ? 2, 0 ? v ? 1 Text Transcription: Double Integral_S (x + y + z) dS x = u + v y = u-v, z = 1 + 2u + v, 0 leqslant u leqslant 2, 0 leqslant v leqslant 1

Evaluate the surface integral. \(\iint_{\mathrm{S}} x y z d S\), \(S\) is the cone with parametric equations \(x=u \cos v\), \(y=u \sin v, z=u, 0 \leqslant u \leqslant 1,0 \leqslant v \leqslant \pi / 2\) Equation Transcription: ?S ???? Text Transcription: double integral _S xyz dS S x=u cos v y=u sin v, z=u, 0 leqslant u leqslant 1, 0 leqslant v leqslant pi/2

Evaluate the surface integral. \(\iint_{S} y d S\), S is the helicoid with vector equation \(\mathbf{r}(u, v)=\langle u \cos v, u \sin v, v\rangle, 0 \leqslant u \leqslant 1,0 \leqslant v \leqslant \pi\)

Evaluate the surface integral. \(\iint_S\left(x^2+y^2\right) d S\) S is the surface with vector equation \(\mathbf{r}(u, v)=\left\langle 2 u v, u^2-v^2, u^2+v^2\right\rangle, u^2+v^2 \leqslant 1\)

Evaluate the surface integral. \(\iint_{s} x^{2} y z d S, \mathrm{S}\) is the part of the plane \(z=1+2 x+3 y\) that lies above the rectangle \([0,3] \times[0,2]\)

Evaluate the surface integral. \(\iint_S x z d S\), S is the part of the plane 2x+2y+z=4 that lies in the first octant

Evaluate the surface integral. \(\iint_{S} x d S\), \(S\) is the triangular region with vertices \((1,0,0),(0,-2,0)\) and \((0,0,4)\) Equation Transcription: Text Transcription: Double integral_s xdS, s (1,0,0),(0,-2,0) and (0,0,4)

Evaluate the surface integral. \(\iint_{s} y d S\), \(S\) is the surface \(z=\frac{2}{3}\left(x^{3 / 2}+y^{3 / 2}\right)\), \(0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1\)

Evaluate the surface integral. \(\iint_{s} z^{2} d S\),S is the part of the paraboloid \(x=y^{2}+z^{2}\) given by \(0 \leqslant x \leqslant 1\) Equation Transcription: ? 0 ? x ? 1 Text Transcription: Double Integral_s z^2 dS x = y^2 + z^2 0 leqslant x leqslant 1

Evaluate the surface integral. \(\iint_{S} y^{2} z^{2} d S\), \(S\) is the part of the cone \(y=\sqrt{x^{2}+z^{2}}\) given by \(0 \leqslant y \leqslant 5\) Equation Transcription: ? 0 ? y ? 5 Text Transcription: Double Integral_s y^2 z^2 dS y = sqrt x^2 + z^2 0 leqslant y leqslant 5

Evaluate the surface integral. \(\iint_{s} x \ d S\), S is the surface \(y=x^{2}+4 z, 0 \leqslant x \leqslant 1,0 \leqslant z \leqslant 1\) Equation Transcription: ? 0 ? x ? 1, 0 ? z ? 1 Text Transcription: Double Integral_s x dS y = x^2 + 4z, 0 leqslant x leqslant 1, 0 leqslant z leqslant 1

Evaluate the surface integral. \(\iint_{S} y^{2} d S\), S is the part of the sphere \(x^{2}+y^{2}+z^{2}=1\) that lies above the cone \(z=\sqrt{x^{2}+y^{2}}\)

Evaluate the surface integral. \(\iint_{s}\left(x^{2} z+y^{2} z\right) \ d S\), S is the hemisphere \(x^{2}+y^{2}+z^{2}=4, \ z \geqslant 0\)

Evaluate the surface integral. \(\iint_{S}(x+y+z) d S\), \(S\) is the part of the half-cylinder \(x^{2}+z^{2}=1, z \geqslant 0\) , that lies between the planes \(y=0\) and \(y=2\) Equation Transcription: ? z ? 0 Text Transcription: Double Integer_s (x + y + z) dS x^2 + z^2 = 1, z geqslant 0 y = 0 y = 2

Evaluate the surface integral. \(\iint_{s} x z \ d S\), \(S\) is the boundary of the region enclosed by the cylinder \(y^{2}+z^{2}=9\) and the planes \(x=0\) and \(x+y=5\) Equation Transcription: ? Text Transcription: Double integral_s xz dS y^2 + z^2 = 9 x = 0 x + y = 5

Evaluate the surface integral. \(\iint_{S}\left(x^{2}+y^{2}+z^{2}\right) d S\), \(S\) is the part of the cylinder \(x^{2}+y^{2}=9\) between the planes \(z=0\) and \(z=2\), together with its top and bottom disks Equation Transcription: ? Text Transcription: Double Integral_s (x^2 + y^2 + z^2) dS x^2 + y^2 = 9 z = 0 z = 2

Evaluate the surface integral \(\iint_{S} F \cdot d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=z e^{x y} i-3 z e^{x y} j+x y k\), \(S\) is the parallelogram of Exercise \(5\) with upward orientation Equation Transcription: Text Transcription: Double integral_s F cdot dS F S F(x,y,z)=ze^xy i - 3ze^xy j + xyk 5

Evaluate the surface integral \(\iint_{S} F \cdot d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=z i+y j+x k\), \(S\) is the helicoid of Exercise \(7\) with upward orientation Equation Transcription: Text Transcription: Double integral_s F cdot dS F S F(x,y,z)=zi+yj+xk 7

Evaluate the surface integral \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. \(\mathbf{F}(x,\ y\ ,z)=xy\ \mathbf{i}+yz\mathbf{\ j}+zx\ \mathbf{k}\), S is the part of the paraboloid \(z=4-x^{2}-y^{2}\) that lies above the square \(0 \leqslant x \leqslant 1, \ 0 \leqslant y \leqslant 1\) , and has upward orientation

Evaluate the surface integral \(\iint_{S} F \cdot \ d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=-x i-y j+z^{3} k\), \(S\) is the part of the cone \(z=\sqrt{x^{2}+y^{2}}\) between the planes \(z = 1\) and \(z = 3\) with downward orientation Equation Transcription: ? Text Transcription: Double Integral_s F cdot dS F(x,y,z) = -xi - yj + z^k z = sqrt x^2 + y^2 z=1 z=3

Evaluate the surface integral \(\iint_S \mathbf{F} \cdot d \mathbf{S}\) for the given vector field \(\mathbf{F}\) and the oriented surface S. In other words, find the flux of \(\mathbf{F}\) across S. For closed surfaces, use the positive (outward) orientation. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z^2 \mathbf{k}\), S is the sphere with radius 1 and center the origin

Evaluate the surface integral \(\iint_{S} F \cdot \ d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=y i-x j+2 z k\), \(S\) is the hemisphere \(x^{2}+y^{2}+z^{2}=4, z \geqslant 0\) , oriented downward Equation Transcription: ? , z ? 0 Text Transcription: Double Integral_s F cdot dS F(x,y,z) = yi - xj + 2zk x^2 + y^2 + z^2 = 4 , z geqslant 0

Evaluate the surface integral \(\iint_{S} F \cdot \ d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=y j-z k\), \(S\) consists of the paraboloid \(y=x^{2}+z^{2}, 0 \leqslant y \leqslant 1\) and the disk \(x^{2}+z^{2} \leqslant 1, y=1\) Equation Transcription: ? , 0 ? y ? 1 ? 1 , Text Transcription: Double Integral_s F times dS F(x,y,z) = yj - zk y = x^2 + z^2, 0 leqslant y leqslant 1 x^2 + z^2 leqslant 1, y = 1

Evaluate the surface integral \(\iint_{S} F \cdot d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=y z i+z x j+x y k, S\) is the surface \(z=x \sin y, 0 \leqslant x \leqslant 2,0 \leqslant y \leqslant \pi\), with upward orientation Equation Transcription: ?S ???? Text Transcription: double integral _S F . dS F S F S F(x,y,z)=yz i + zx j + xy k, S z=xsiny, 0 leqslant x leqslant 2, 0 leqslant y leqslant pi

Evaluate the surface integral \(\iint_{S} F \cdot \ d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=x i+2 y j+3 z k\), S is the cube with vertices \((\pm 1, \pm 1, \pm 1)\) Equation Transcription: ? Text Transcription: Double Integer_s F times dS F(x,y,z) = xi + 2yj + 3zk (pm 1, pm 1, pm 1)

Evaluate the surface integral \(\iint_S \mathbf{F} \cdot d \mathbf{S}\) for the given vector field \(\mathbf{F}\) and the oriented surface S. In other words, find the flux of \(\mathbf{F}\) across S. For closed surfaces, use the positive (outward) orientation. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+5 \mathbf{k}\), S is the boundary of the region enclosed by the cylinder \(x^2+z^2=1\) and the planes y=0 and x+y=2

Evaluate the surface integral \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. \(\mathbf{F}(x,\ y,\ z)=x^{2\ }\mathbf{i}+y^2\mathbf{\ j}+z^2\mathbf{\ k}\), S is the boundary of the solid half-cylinder \(0 \leqslant z \leqslant \sqrt{1-y^{2}}, \ 0 \leqslant x \leqslant 2\)

Evaluate the surface integral \(\iint_{S} F \cdot d S\) for the given vector field \(F\) and the oriented surface \(S\). In other words, find the flux of \(F\) across \(S\). For closed surfaces, use the positive (outward) orientation. \(F(x, y, z)=y i+(z-y) j+x k\),\(S\) is the surface of the tetrahedron with vertices \((0,0,0),(1,0,0),(0,1,0)\), and \((0,0,1)\) Equation Transcription: Text Transcription: Double integral_s F cdot dS F S F(x,y,z)=yi+(z-y)j+xk (0,0,0), (1,0,0), (0,1,0), and (0,0,1)

Use a computer algebra system to evaluate \(\iint_{S}\left(x^{2}+y^{2}+z^{2}\right) \ d S\) correct to four decimal places, where S is the surface \(z=x e^{y}, 0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1\) Equation Transcription: ? , 0 ? x ? 1, 0 ? y ? 1 Text Transcription: Double Integer_s (x^2 + y^2 + z^2) dS z = xe^y, 0 leqslant x leqslant 1, 0 leqslant y leqslant 1

Use a computer algebra system to find the exact value of \(\iint_{s} x y z \ d S\), where S is the surface \(z=x^{2} y^{2}, 0 \leqslant \mathrm{x} \leqslant 1,0 \leqslant \mathrm{y} \leqslant 2\) Equation Transcription: ? , 0 ? x ? 1, 0 ? y ? 2 Text Transcription: Double Integer_s xyz dS z = x^2 y^2, 0 leqslant x leqslant 1, 0 leqslant y leqslant 2

Use a computer algebra system to find the value of \(\iint_{S} x^{2} y^{2} z^{2} \ d S\) correct to four decimal places, where S is the part of the paraboloid \(z=3-2 x^{2}-y^{2}\) that lies above the xy-plane. Equation Transcription: ? Text Transcription: Double Integral_s x^2 y^2 z^2 dS z = 3 - 2x^2 - y^2

Use a computer algebra system to find the flux of \(F(x, y, z)=\sin (x y z) i+x^{2} y j+z^{2} e^{x / 5} k\) across the part of the cylinder \(4 y^{2}+z^{2}=4\) that lies above the \(x y\) -plane and between the planes \(x=-2\) and \(x=2\) with upward orientation. Illustrate by graphing the cylinder and the vector field on the same screen. Equation Transcription: Text Transcription: F(x,y,z)=sin(xyz)i + x^2yj + z^2 e^x/5 k 4y^2 + z^2 = 4 xy x=-2 x=2

Find a formula for \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) similar to Formula 10 for the case where S is given by \(y=h(x, \ z)\) and n is the unit normal that points toward the left (when the axes are drawn in the usual way).

Find a formula for \(\iint_S \mathbf{F} \cdot d \mathbf{S}\) similar to Formula 10 for the case where S is given by x=k(y, z) and \(\mathbf{n}\) is the unit normal that points forward (that is, toward the viewer when the axes are drawn in the usual way).

Find the center of mass of the hemisphere \(x^{2}+y^{2}+z^{2}=a^{2}, z \geqslant 0\), if it has constant density. Equation Transcription: Text Transcription: x^2+y^2+z^2=a^2, z geqslant 0

Find the mass of a thin funnel in the shape of a cone \(z=\sqrt{x^{2}+y^{2}}, 1 \leqslant z \leqslant 4\), if its density function is \(\rho(x, y, z)=10-z\) Equation Transcription: Text Transcription: z= sqrt x^2+y^2, 1 leqslant z leqslant 4 rho(x,y,z)=10-z

(a) Give an integral expression for the moment of inertia \(I_{z}\) about the \(z\) -axis of a thin sheet in the shape of a surface \(S\) if the density function is \(\rho\). (b) Find the moment of inertia about the \(z\) -axis of the funnel in Exercise \(40\) . Equation Transcription: Text Transcription: I_z z S rho 40

Let \(S\) be the part of the sphere \(x^{2}+y^{2}+z^{2}=25\) that lies above the plane \(z=4\). If \(S\) has constant density \(k\), find (a) the center of mass and (b) the moment of inertia about the \(z\) -axis. Equation Transcription: Text Transcription: S x^2+y^2+z^2=25 z=4 k z

A fluid has density \(870 \mathrm{~kg} / \mathrm{m}^{3}\) and flows with velocity \(v=z i+y^{2} j+x^{2} k\), where \(x,y\), and \(z\) are measured in meters and the components of \(v\) in meters per second. Find the rate of flow outward through the cylinder \(x^{2}+y^{2}=4,0 \leqslant z \leqslant 1\) Equation Transcription: Text Transcription: 870 kg/m^3 v=zi+y^2 j+x2k x,y, and z v x^2+y^2=4, 0 leqslant z leqslant 1

Seawater has density \(1025 \mathrm{~kg} / \mathrm{m}^{3}\) and flows in a velocity field \(v=y i+x j\), where \(x,y\), and \(z\) are measured in meters and the components of \(v\) in meters per second. Find the rate of flow outward through the hemisphere \(x^{2}+y^{2}+z^{2}=9, z \geqslant 0\). Equation Transcription: Text Transcription: 1025 kg/m^3 v=yi+xj v x^2+y^2+z^2=9, z geqslant 0

Use Gauss's Law to find the charge contained in the solid hemisphere \(x^{2}+y^{2}+z^{2} \leqslant a^{2}, z \geqslant 0\), if the electric field is \(E(x, y, z)=x i+y j+2 z k\) Equation Transcription: Text Transcription: x^2+y^2+z^2 leqslant a^2, z geqslant 0 E(x,y,z)=xi+yj+2zk

Use Gauss's Law to find the charge enclosed by the cube with vertices \((\pm 1, \pm 1, \pm 1)\) if the electric field is \(E(x, y, z)=x i+y j+z k\) Equation Transcription: Text Transcription: (pm 1, pm 1, pm 1) E(x,y,z)=xi+yj+zk

The temperature at the point (x, y, z) in a substance with conductivity K=6.5 is \(u(x, y, z)=2 y^2+2 z^2\). Find the rate of heat flow inward across the cylindrical surface \(y^2+z^2=6,0 \leqslant x \leqslant 4\)

The temperature at a point in a ball with conductivity \(K\) is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere \(S\) of radius \(a\) with center at the center of the ball. Equation Transcription: Text Transcription: K S a

Let \(F\) be an inverse square field, that is, \(F(\mathbf{r})=\mathrm{cr} /|\mathbf{r}|^{3}\) for some constant \(c\), where \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Show that the flux of \(F\) across a sphere \(S\) with center the origin is independent of the radius of \(S\). Equation Transcription: Text Transcription: F F(r)=cr/|r|^3 c r=xi+yj+zk S