Moment of Inertia In Exercises 35 and 36, find for the given lamina with uniform density

In Exercises 1 - 4, evaluate \(\int_{S} \int(x-2 y+z) d S\). \(S: z=4-x, \quad 0 \leq x \leq 4, \quad 0 \leq y \leq 3\) Text Transcription: int_S int(x - 2y + z) dS S: z = 4 - x, 0 leq x leq 4, 0 leq y leq 3

In Exercises 1 - 4, evaluate \(\int_{S} \int(x-2 y+z) d S\). \(S: z=15-2 x+3 y, \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 4\) Text Transcription: int_S int(x - 2y + z) dS S: z = 15 - 2x + 3y, 0 leq x leq 2, 0 leq y leq 4

In Exercises 1 - 4, evaluate \(\int_{S} \int(x-2 y+z) d S\). \(S: z=2, x^{2}+y^{2} \leq 1\) Text Transcription: int_S int(x - 2y + z) dS S: z = 2, x^2 + y^2 leq 1

In Exercises 1 - 4, evaluate \(\int_{S} \int(x-2 y+z) d S\). \(S: z=\frac{2}{3} x^{3 / 2}, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq x\) Text Transcription: int_S int(x - 2y + z) dS S: z = 2 / 3 x^{3/2}, 0 leq x leq 1, 0 leq y leq x

In Exercises 5 and 6, evaluate \(\int_{S} \int x y d S\). S: z = 3 - x - y, first octant Text Transcription: int_S int xy dS

In Exercises 5 and 6, evaluate \(\int_{S} \int x y d S\). \(S: z=h, \quad 0 \leq x \leq 2, \quad 0 \leq y \leq \sqrt{4-x^{2}}\) Text Transcription: int_S int xy dS S: z = h, 0 leq x leq 2, 0 leq y leq sqrt{4 - x^2}

In Exercises 7 and 8, use a computer algebra system to evaluate \(\int_{S} \int x y d S\) \(S: z=9-x^{2}, \quad 0 \leq x \leq 2, \quad 0 \leq y \leq x\) Text Transcription: int_S int xy dS S: z = 9 - x^2, 0 leq x leq 2, 0 leq y leq x

In Exercises 7 and 8, use a computer algebra system to evaluate \(\int_{S} \int x y d S\) \(S: z=\frac{1}{2} x y, \quad 0 \leq x \leq 4, \quad 0 \leq y \leq 4\) Text Transcription: int_S int xy dS S: z = 1 / 2 xy, 0 leq x leq 4, 0 leq y leq 4

In Exercises 9 and 10, use a computer algebra system to evaluate \(int_{S} \int\left(x^{2}-2 x y\right) d S\) \(S: z=10-x^{2}-y^{2}, \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 2\) Text Transcription: int_S int(x^2 - 2xy) dS S: z = 10 - x^2 - y^2, 0 leq x leq 2, 0 leq y leq 2

In Exercises 9 and 10, use a computer algebra system to evaluate \(int_{S} \int\left(x^{2}-2 x y\right) d S\) \(S: z=\cos x, \quad 0 \leq x \leq \frac{\pi}{2}, \quad 0 \leq y \leq \frac{1}{2} x\) Text Transcription: int_S int(x^2 - 2xy) dS S: z = cos x, 0 leq x leq pi / 2, 0 leq y leq 1 / 2 x

In Exercises 11 and 12, find the mass of the surface lamina S of density \(\rho\). S: 2x + 3y + 6z = 12, first octant, \(\rho(x, y, z)=x^{2}+y^{2}\) Text Transcription: rho rho(x, y, z) = x^2 + y^2

In Exercises 11 and 12, find the mass of the surface lamina S of density \(\rho\). \(S: z=\sqrt{a^{2}-x^{2}-y^{2}}, \quad \rho(x, y, z)=k z\) Text Transcription: rho S: z = sqrt{a^2 - x^2 - y^2}, rho(x, y, z) = kz

In Exercises 13 - 16, evaluate \(\int_{S} \int f(x, y) d S\). f(x, y) = y + 5 \(S: \mathbf{r}(u, v)=u \mathbf{i}+v \mathbf{j}+2 v \mathbf{k}, \quad 0 \leq u \leq 1, \quad 0 \leq v \leq 2\) Text Transcription: int_S int f(x, y) dS S: r(u, v) = ui + vj + 2vk, 0 leq u leq 1, 0 leq v leq 2

In Exercises 13 - 16, evaluate \(\int_{S} \int f(x, y) d S\). f(x, y) = xy \(S: \mathbf{r}(u, v)=2 \cos u \mathbf{i}+2 \sin u \mathbf{j}+v \mathbf{k}\) \(0 \leq u \leq \frac{\pi}{2}, \quad 0 \leq v \leq 1\) Text Transcription: int_S int f(x, y) dS S: r(u, v) = 2 cos ui + 2 sin uj + vk 0 leq u leq pi / 2, 0 leq v leq 1

In Exercises 13 - 16, evaluate \(\int_{S} \int f(x, y) d S\). f(x, y) = x + y \(S: \mathbf{r}(u, v)=2 \cos u \mathbf{i}+2 \sin u \mathbf{j}+v \mathbf{k}\) \(0 \leq u \leq \frac{\pi}{2}, \quad 0 \leq v \leq 1\) Text Transcription: int_S int f(x, y) dS S: r(u, v) = 2 cos ui + 2 sin uj + vk 0 leq u leq pi / 2, 0 leq v leq 1

In Exercises 13 - 16, evaluate \(\int_{S} \int f(x, y) d S\). f(x, y) = x + y \(S: \mathbf{r}(u, v)=4 u \cos v \mathbf{i}+4 u \sin v \mathbf{j}+3 u \mathbf{k}\) \(0 \leq u \leq 4, \quad 0 \leq v \leq \pi\) Text Transcription: int_S int f(x, y) dS S: r(u, v) = 4u cos vi + 4u sin vj + 3uk 0 leq u leq 4, 0 leq v leq pi

In Exercises 17 – 22, evaluate \(\int_{S} \int f(x, y, z) d S\). \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) \(S: z=x+y, \quad x^{2}+y^{2} \leq 1\) Text Transcription: int_S int f(x, y, z) dS f(x, y, z) = x^2 + y^2 + z^2 S: z = x + y, x^2 + y^2 leq 1

In Exercises 17 – 22, evaluate \(\int_{S} \int f(x, y, z) d S\). \(f(x, y, z)=\frac{x y}{z}\) \(S: z=x^{2}+y^{2}, \quad 4 \leq x^{2}+y^{2} \leq 16\) Text Transcription: int_S int f(x, y, z) dS f(x, y, z) = xy / z S: z = x^2 + y^2, 4 leq x^2 + y^2 leq 16

In Exercises 17 – 22, evaluate \(\int_{S} \int f(x, y, z) d S\). \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\) \(S: z=\sqrt{x^{2}+y^{2}}, \quad x^{2}+y^{2} \leq 4\) Text Transcription: int_S int f(x, y, z) dS f(x, y, z) = sqrt{x^2 + y^2 + z^2} S: z = sqrt{x^2 + y^2}, x^2 + y^2 leq 4

In Exercises 17 – 22, evaluate \(\int_{S} \int f(x, y, z) d S\). \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\) \(S: z=\sqrt{x^{2}+y^{2}}, \quad(x-1)^{2}+y^{2} \leq 1\) Text Transcription: int_S int f(x, y, z) dS f(x, y, z) = sqrt{x^2 + y^2 + z^2} S: z = sqrt{x^2 + y^2}, (x - 1)^2 + y^2 leq 1

In Exercises 17 – 22, evaluate \(\int_{S} \int f(x, y, z) d S\). \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) \(S: x^{2}+y^{2}=9, \quad 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad 0 \leq z \leq 9\) Text Transcription: int_S int f(x, y, z) dS f(x, y, z) = x^2 + y^2 + z^2 S: x^2 + y^2 = 9, 0 leq x leq 3, 0 leq y leq 3, 0 leq z leq 9

In Exercises 17 – 22, evaluate \(\int_{S} \int f(x, y, z) d S\). \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) \(S: x^{2}+y^{2}=9, \quad 0 \leq x \leq 3, \quad 0 \leq z \leq x\) Text Transcription: int_S int f(x, y, z) dS f(x, y, z) = x^2 + y^2 + z^2 S: x^2 + y^2 = 9, 0 leq x leq 3, 0 leq z leq x

In Exercises 23 - 28, find the flux of F through S, \(\int_{S} \int F \cdot N d S\) where N is the upward unit normal vector to S. \(\mathbf{F}(x, y, z)=3 z \mathbf{i}-4 \mathbf{j}+y \mathbf{k}\) S: z = 1 - x - y, first octant Text Transcription: int_S int F cdot N dS F(x, y, z) = 3zi - 4j + yk

In Exercises 23 - 28, find the flux of F through S, \(\int_{S} \int F \cdot N d S\) where N is the upward unit normal vector to S. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}\) S: z = 6 - 3x - 2y, first octant Text Transcription: int_S int F cdot N dS F(x, y, z) = xi + yj

In Exercises 23 - 28, find the flux of F through S, \(\int_{S} \int F \cdot N d S\) where N is the upward unit normal vector to S. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) \(S: z=1-x^{2}-y^{2}, \quad z \geq 0\) Text Transcription: int_S int F cdot N dS F(x, y, z) = xi + yj + zk S: z = 1 - x^2 - y^2, z geq 0

In Exercises 23 - 28, find the flux of F through S, \(\int_{S} \int F \cdot N d S\) where N is the upward unit normal vector to S. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) \(S: x^{2}+y^{2}+z^{2}=36\), first octant Text Transcription: int_S int F cdot N dS F(x, y, z) = xi + yj + zk S: x^2 + y^2 + z^2 = 36

In Exercises 23 - 28, find the flux of F through S, \(\int_{S} \int F \cdot N d S\) where N is the upward unit normal vector to S. \(\mathbf{F}(x, y, z)=4 \mathbf{i}-3 \mathbf{j}+5 \mathbf{k}\) \(S: z=x^{2}+y^{2}, \quad x^{2}+y^{2} \leq 4\) Text Transcription: int_S int F cdot N dS F(x, y, z) = 4i - 3j + 5k S: z = x^2 + y^2, x^2 + y^2 leq 4

In Exercises 23 - 28, find the flux of F through S, \(\int_{S} \int F \cdot N d S\) where N is the upward unit normal vector to S. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}-2 z \mathbf{k}$\) \(S: z=\sqrt{a^{2}-x^{2}-y^{2}}\) Text Transcription: int_S int F cdot N dS F(x, y, z) = xi + yj - 2zk S: z = sqrt{a^2 - x^2 - y^2}

In Exercises 29 and 30, find the flux of F over the closed surface. (Let N be the outward unit normal vector of the surface.) \(\mathbf{F}(x, y, z)=(x+y) \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) \(S: z=16-x^{2}-y^{2}, \quad z=0\) Text Transcription: F(x, y, z) = (x + y)i + yj + zk S: z = 16 - x^2 - y^2, z = 0

In Exercises 29 and 30, find the flux of F over the closed surface. (Let N be the outward unit normal vector of the surface.) \(\mathbf{F}(x, y, z)=4 x y \mathbf{i}+z^{2} \mathbf{j}+y z \mathbf{k}\) S: unit cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1 Text Transcription: F(x, y, z) = 4xyi + z^{2}j + yzk

Let \(\mathbf{E}=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}\) be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\) and its circular base in the xy-plane. Text Transcription: E = yzi + xzj + xyk z = sqrt{1 - x^2 - y^2}

Let \(\mathbf{E}=x \mathbf{i}+y \mathbf{j}+2 z \mathbf{k}\) be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\) and its circular base in the xy-plane. Text Transcription: E = xi + yj + 2zk z = sqrt{1 - x^2 = -y^2

In Exercises 33 and 34, use the following formulas for the moments of inertia about the coordinate axes of a surface lamina of density \(\rho\). \(I_{x}=\int_{S} \int\left(y^{2}+z^{2}\right) \rho(x, y, z) d S\) \(I_{y}=\int_{S} \int\left(x^{2}+z^{2}\right) \rho(x, y, z) d S\) \(I_{z}=\int_{S} \int\left(x^{2}+y^{2}\right) \rho(x, y, z) d S\) Verify that the moment of inertia of a conical shell of uniform density about its axis is \(\frac{1}{2} m a^{2}\), where m is the mass and a is the radius and height. Text Transcription: rho I_x = int_S int(y^2 + z^2) rho(x, y, z) dS I_y = int_S int(x^2 + z^2) rho(x, y, z) dS I_z = int_S int(x^{2} + y^2) rho(x, y, z) dS 1 / 2 ma^2

In Exercises 33 and 34, use the following formulas for the moments of inertia about the coordinate axes of a surface lamina of density \(\rho\). \(I_{x}=\int_{S} \int\left(y^{2}+z^{2}\right) \rho(x, y, z) d S\) \(I_{y}=\int_{S} \int\left(x^{2}+z^{2}\right) \rho(x, y, z) d S\) \(I_{z}=\int_{S} \int\left(x^{2}+y^{2}\right) \rho(x, y, z) d S\) Verify that the moment of inertia of a spherical shell of uniform density about its diameter is \(\frac{2}{3} m a^{2}\), where m is the mass and a is the radius. Text Transcription: rho I_x = int_S int(y^2 + z^2) rho(x, y, z) dS I_y = int_S int(x^2 + z^2) rho(x, y, z) dS I_z = int_S int(x^{2} + y^2) rho(x, y, z) dS 2 / 3 ma^2

In Exercises 35 and 36, find \(I_{z}\) for the given lamina with a uniform density of 1 . Use a computer algebra system to verify your results. \(x^{2}+y^{2}=a^{2}, \quad 0 \leq z \leq h\) Text Transcription: I_z x^2 + y^2 = a^2, 0 leq z leq h

In Exercises 35 and 36, find \(I_{z}\) for the given lamina with a uniform density of 1 . Use a computer algebra system to verify your results. \(z=x^{2}+y^{2}, \quad 0 \leq z \leq h\) Text Transcription: I_z z = x^2 + y^2, 0 leq z leq h

In Exercises 37 and 38, use a computer algebra system to find the rate of mass flow of a fluid of density \(\rho\) through the surface S oriented upward when the velocity field is given by F(x, y, z) = 0.5zk. \(S: z=16-x^{2}-y^{2}, \quad z \geq 0\) Text Transcription: rho S: z = 16 - x^2 - y^2, z geq 0

In Exercises 37 and 38, use a computer algebra system to find the rate of mass flow of a fluid of density \(\rho\) through the surface S oriented upward when the velocity field is given by F(x, y, z) = 0.5zk. \(S: z=\sqrt{16-x^{2}-y^{2}}\) Text Transcription: rho S: z = sqrt{16 - x^2 - y^2}

Define a surface integral of the scalar function f over a surface z = g(x, y). Explain how to evaluate the surface integral.

Orientable Surface Describe an orientable surface.

Flux Integral Define a flux integral and explain how it is evaluated.

Is the surface shown in the figure orientable? Explain why or why not.

Investigation (a) Use a computer algebra system to graph the vector-valued function \(\mathbf{r}(u, v)=(4-v \sin u) \cos (2 u) \mathbf{i}+(4-v \sin u) \sin (2 u) \mathbf{j}+\) \(v \cos u \mathbf{k}, \quad 0 \leq u \leq \pi, \quad-1 \leq v \leq 1\) This surface is called a Möbius strip. (b) Is the surface orientable? Explain why or why not. (c) Use a computer algebra system to graph the space curve represented by r(u, 0). Identify the curve. (d) Construct a Möbius strip by cutting a strip of paper, making a single twist, and pasting the ends together. (e) Cut the Möbius strip along the space curve graphed in part (c), and describe the result. Text Transcription: r(u, v) = (4 - v sin u) cos (2u)i + (4 - v sin u) sin (2u)j + v cos uk, 0 leq u leq pi, -1 leq v leq 1