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15.7 Assess Your Understanding

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Concepts and Vocabulary

  1. Multiple Choice If a surface S has a unit normal vector n at each point of the surface that can be chosen so n varies continuously over the surface, and as n moves around any closed curve on the surface, n returns to its original direction, then the surface is said to be [(a) a Möbius strip, (b) integrable, (c) orientable].

(c)

  1. Multiple Choice  If F=F(x,y,z) is a vector field that represents the velocity of a fluid, and ρ=ρ(x,y,z) is the variable mass density of the fluid, then the surface integral ∬ is called the [(a) speed, (b) flux, (c) orientation, (d) acceleration] of \mathbf{F} across the surface S.

(b)

Skill Building

In Problems 3–12, find each surface integral.

  1. \iint\limits_{\kern-3ptS}(x^{2}+y^{2})\,dS;\quad S: z=f(x,y)=2x+3y+5, \ 0\leq x\leq 1, 0\leq y\leq 1

\displaystyle{2 \over 3}\sqrt {14}

  1. \iint\limits_{\kern-3ptS}x^{2}y^{2}\,dS;\quad S: z=f(x,y)=3x+4y+8, 0\leq x\leq 1, 0\leq y\leq 1

  1. \iint\limits_{\kern-3ptS}y\,dS;\quad S: z=f(x,y)=x+5y^{2}, 0\leq x\leq2, 0\leq y\leq1

\displaystyle{{17} \over {25}}\sqrt {102} - \displaystyle{1 \over {75}}\sqrt 2

  1. \iint\limits_{\kern-3ptS}4x\,dS;\quad S: z=f(x,y)=2x^{2}+y, 0\leq x\leq1, 0\leq y\leq2

  1. \iint\limits_{\kern-3ptS}(x+y+z)\,dS, where S is the portion of the plane z=x-3y above the region enclosed by y=0, x=1, and x=3y in the xy-plane

\displaystyle{5 \over {27}} \sqrt {11}

  1. \iint\limits_{\kern-3ptS}x\,dS, where S is the portion of the plane x+y+2z=4 above the region enclosed by x=1, y=1, x=0, and y=0

  1. \iint\limits_{\kern-3ptS}y\,dS, where S is the portion of the plane x+y+z=2 inside the cylinder x^{2}+y^{2}=1

0

  1. \iint\limits_{\kern-3ptS}yz\,dS, where S is the portion of the plane x+2y+3z=6 in the first octant

  1. \iint\limits_{\kern-3ptS}x^{2}z\,dS, where S is the surface x^{2}+y^{2}=1, 0\leq z\leq 1

\displaystyle{\pi \over 2}

  1. \iint\limits_{\kern-3ptS}(x+y)z\,dS, where S is the surface x^{2} + y^{2} = 9, 1\leq z \leq 3

In Problems 13–16, find the outer unit normal vector to the surface S defined by z=f(x,y) .

  1. z=f(x,y)=\sqrt{16-x^{2}-y^{2}},\quad 0\leq x^{2}+y^{2}\leq 16

\displaystyle{x \over 4} {{\bf i}} + \displaystyle{y \over 4} {{\bf j}} + \displaystyle{{\sqrt {16 - x^2 - y^2} } \over 4} {{\bf k}}

  1. z=f(x,y)=\sqrt{1-x^{2}-y^{2}},\quad 0\leq x^{2}+y^{2}\leq 1

  1. z=f(x,y)=\sqrt{36-9x^{2}-4y^{2}},\quad 0\leq 9x^{2}+4y^{2}\leq 36

\displaystyle{{9x} \over {2\sqrt {9 + 18x^2 + 3y^2} }} {{\bf i}} + \displaystyle{{4y} \over {2\sqrt {9 + 18x^2 + 3y^2} }} {{\bf j}} + \displaystyle{{\sqrt {36 - 9x^2 - 4y^2} } \over {2\sqrt {9 + 18x^2 + 3y^2} }} {{\bf k}}

  1. z=f(x,y)=\sqrt{4-x^{2}-4y^{2}},\quad 0\leq x^{2}+4y^{2}\leq 4

In Problems 17 and 18, find both orientations for each surface.

  1. S: \mathbf{r}(u,v)=(u+v)\mathbf{i}+u^{2}\mathbf{j}+v^{2}\mathbf{k}

{{\bf n}} = \displaystyle{{4uv} \over {2\sqrt {4u^2v^2 + v^2 + u^2} }} {{\bf i}} + \displaystyle{{ - 2v} \over {2\sqrt {4u^2v^2 + v^2 + u^2} }} {{\bf j}} + \displaystyle{{ - 2u} \over {2\sqrt {4u^2v^2 + v^2 + u^2} }} {{\bf k}} -{{\bf n}} = \displaystyle{{ - 4uv} \over {2\sqrt {4u^2v^2 + v^2 + u^2} }} {{\bf i}} + \displaystyle{{2v} \over {2\sqrt {4u^2v^2 + v^2 + u^2} }} {{\bf j}} + \displaystyle{{2u} \over {2\sqrt {4u^2v^2 + v^2 + u^2} }} {{\bf k}}

  1. S: \mathbf{r}(u,v)=uv\mathbf{i}+v\mathbf{j}+e^{u}\mathbf{k}

Flux Across a Surface  In Problems 19–26, a fluid with constant mass density \rho flows across the cube shown in Figure 66. If the velocity of the fluid at any point on the cube is given by \mathbf{F}, find the flux of \mathbf{F} across the cube in the direction of the outer unit normal vectors.

  1. \mathbf{F}=x\mathbf{i}

\rho

  1. \mathbf{F}=y\mathbf{i}

  1. \mathbf{F}=z\mathbf{i}

0

  1. \mathbf{F}=x\mathbf{i}+y\mathbf{j}

  1. \mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}

3\rho

  1. \mathbf{F}=z^{2}\mathbf{i}

  1. \mathbf{F}=x^{2}\mathbf{i}+y^{2}\mathbf{j}+z^{2}\mathbf{k}

3\rho

  1. \mathbf{F}=x^{2}\mathbf{i}+y^{2}\mathbf{j}

Applications and Extensions

Flux Across a Surface In Problems 27–30, find \iint\limits_{\kern-3ptS}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS, where \mathbf{n} is the outer unit normal of S.

  1. \mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}, and S is the surface x^{2}+y^{2}+z^{2}=1, z\geq 0

2\pi

  1. \mathbf{F}=-y\mathbf{i}+x\mathbf{j}+z\mathbf{k}, and S is the surface x^{2}+y^{2}+z^{2}=1, z\geq 0

1037

  1. \mathbf{F}=(x+y)\mathbf{i}+(2x-z)\mathbf{j}+y\mathbf{k}, and S is the tetrahedron formed by the coordinate planes and the plane z+2x+2y=8

\displaystyle{{160} \over 3}

  1. \mathbf{F}=2x\mathbf{i}-x^{2}\mathbf{j}+(z-2x+2y)\mathbf{k} , and S is the tetrahedron formed by the coordinate planes and the plane 2x+2y+z=6

In Problems 31 and 32, find the flux integral \iint\limits_{\kern-3ptS} \mathbf{F}\,{\cdot}\, \mathbf{n}\,dS, given the oriented surface S and vector field \mathbf{F}.

  1. The surface S is parametrized by \mathbf{r}(u,v)=uv\,\mathbf{i}+u^{2}\mathbf{j}+v\,\mathbf{k}, 1\leq u\leq 2, 0\leq v\leq 3 and the vector field is \mathbf{F}=\mathbf{F}(x,y,z)=-z\mathbf{j}+\mathbf{k}.

-5

  1. The surface S is parametrized by \mathbf{r}(u,v)=ve^{u} \mathbf{i}+v^{3}\mathbf{j}+u\mathbf{k}, 0\leq u\leq 1, -1\leq v\leq 1 and the vector field is \mathbf{F}=\mathbf{F}(x,y,z)=xy\mathbf{i}+2z\mathbf{ j}.

  1. Mass of a Lamina Find the mass of a lamina in the shape of a cone z=\sqrt{x^{2}+y^{2}}, 2\leq z\leq 4, if the mass density of the lamina is \rho =\rho (x,y,z)=8-z.

\displaystyle{{176} \over 3}\sqrt 2 \pi

  1. Mass of a Lamina Find the mass of a lamina in the shape of a cone z=\sqrt{x^{2}+y^{2}}, 2\leq z\leq 4, if the mass density of the lamina is \rho =\rho (x,y,z)=\sqrt{x^{2}+y^{2}}.

  1. Center of Mass of a Lamina Find the center of mass of a lamina in the shape of the part of the plane 2x+3y+z=4 that lies inside x^{2}+z^{2}=9, if the mass density of the lamina is \rho =\rho (x,y,z)= \sqrt{x^{2}+z^{2}}.

(\bar {x},\bar {y},\bar {z}) = \left(0,\displaystyle{4 \over 3},0\right)

  1. Center of Mass of a Lamina Find the center of mass of a lamina in the shape of the unit sphere x^{2}+y^{2}+z^{2}=1, if the mass density of the lamina is \rho =\rho (x,y,z)=z+1.

In Problems 37 and 38, find each surface integral.

  1. \iint\limits_{\kern-3ptS}x^{2}z\,dS;\quad S: the portion of the cylinder x^{2}+y^{2}=9 between z=0 and z=5

\displaystyle{{675} \over 2}\pi

  1. \iint\limits_{\kern-3ptS}(x+2y)\,dS ;\quad S: \mathbf{r}(u,v)=u\cos v\,\mathbf{i}+(u^{2}+1)\mathbf{j}+u\cos v\,\mathbf{k}, 0\leq u\leq 1 and 0\leq v\leq \pi /2

  1. Flux Across a Surface Find the mass of a fluid with constant mass density flowing across the paraboloid z=x^{2}+y^{2}, z\leq 5 , in a unit of time in the direction of the outer unit normal, if the velocity of the fluid at any point on the paraboloid is \mathbf{F}=\mathbf{F }(x,y,z)=-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}.

\displaystyle{{25} \over 2}\pi \rho

  1. Flux Across a Surface  Find the mass of a fluid with constant mass density flowing across the paraboloid z=9-x^{2}-y^{2}, z\geq 0, in a unit of time in the direction of the outer unit normal, if the velocity of the fluid at any point on the paraboloid is \mathbf{F}=\mathbf{F} (x,y,z)=x\mathbf{i}+y\mathbf{j}+3\mathbf{k}.

  1. If \mathbf{n} is the upward-pointing unit normal to a surface z=f(x,y) and \mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}, show that \iint\limits_{\kern-3ptS}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS=\iint\limits_{\kern-3ptD}\left( -P \dfrac{\partial f}{\partial x}-Q\dfrac{\partial f}{\partial y}+R\right) dx\,dy

See Student Solutions Manual.

  1. Electric Flux Find the electric flux across the part of the cylinder x^{2}+y^{2}=9, 0\leq z\leq 2, in the direction of the outer unit normal vectors when the electric field is \mathbf{E}(x,y,z)=x\mathbf{i} +2y\mathbf{j}+3z\mathbf{k}.

  1. Helicoid Find \iint\limits_{\kern-3ptS}(x^2+z^2)dS, where S is the helicoid \mathbf{r}(u,v)=u\cos(2v){\bf i} + u\sin (2v){\bf j} + v{\bf k},0\leq u\leq 3, 0\leq v\leq 2\pi.

- \displaystyle{1 \over {32}}\pi \ln 2 - \displaystyle{1 \over {64}}\pi \ln 3 + \displaystyle{{219} \over {32}}\pi \sqrt {37} - \displaystyle{1 \over {64}}\pi \ln \left(\displaystyle{1 \over 2} + \displaystyle{1 \over {12}}\sqrt {37} \right) + \displaystyle{4 \over 3}\pi ^3\ln 2 + \displaystyle{2 \over 3}\pi ^3\ln 3 + 4\pi ^3\sqrt {37} + \displaystyle{2 \over 3}\pi ^3\ln \left(\displaystyle{1 \over 2} + \displaystyle{1 \over {12}}\sqrt {37} \right) \approx 936.58

Challenge Problems

  1. Center of Mass Find the center of mass of the upper half of the sphere x^{2}+y^{2}+z^{2}=a^{2} covered by a thin material with mass density at each point proportional to the distance from the xy-plane.

  1. Find \iint\limits_{\kern-3ptS}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS, where S is a level surface of a function w= f(x,y,z) with outer unit normal vector \mathbf{n}, \ \mathbf{F}={\nabla} f, and f satisfies the equation \left( \dfrac{\partial f}{\partial x}\right) ^{2}+\left( \dfrac{\partial f}{ \partial y}\right) ^{2}+\left( \dfrac{\partial f}{\partial z}\right) ^{2}=1. ( Hint: The answer should depend on S.)

\int\!\!\int\limits_{S} {{{\bf F}} \cdot {{\bf n}} dS} is equal to the surface area of S.