Answer: Mass and center of mass Let S be a surface that

Describe the surface with the given parametric representation. \(\mathbf{r}(u, v)=\langle u, v, 2 u+3 v-1\rangle\), for \(\leq u \leq 3,2 \leq v \leq 4\)

Find the area of the following surfaces using a parametric description of the surface. The hemisphere \(x^{2}+y^{2}+z^{2}=100\), for \(z \geq 0\)

Describe the surface with the given parametric representation. \(\mathbf{r}(u, v)=\langle v, 6 \cos u, 6 \sin u\rangle\), for \(0 \leq u \leq 2 \pi, 0 \leq v \leq 2\)

Find the area of the following surfaces using a parametric description of the surface. The plane \(z=3-x-3 y\) in the first octant

Find the area of the following surfaces using a parametric description of the surface. The half cylinder \(\{(r, \theta, z): r=4,0 \leq \theta \leq \pi, 0 \leq z \leq 7\}\)

Find the average value of the temperature function T(x, y, z) = 100 - 25z on the cone \(z^{2}=x^{2}+y^{2}\), for \(0 \leq z \leq 2\).

Consider the radial fields \(\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}\), where p is a real number. Let S consist of the spheres A and B centered at the origin with radii \(0<a<b\), respectively. The total outward flux across S consists of the outward flux across the outer sphere B minus the inward flux across the inner sphere A. a. Find the total flux across S with p = 0. Interpret the result. b. Show that for p = 3 (an inverse square law), the flux across S is independent of a and b.

Let S be a surface that represents a thin shell with density p. The moments about the coordinate planes (see Section 13.6) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S\), \(M_{x z}=\iint_{S} y \rho(x, y, z) d S, M_{x y}=\iint_{S} z \rho(x, y, z) d S\). The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}\), \(\bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m}\), where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The constant-density hemispherical shell \(x^{2}+y^{2}+z^{2}=a^{2}, z \geq 0\)

Describe the surface with the given parametric representation. \(\mathbf{r}(u, v)=\langle v \cos u, v \sin u, 4 v\rangle\), for \(0 \leq u \leq \pi, 0 \leq v \leq 3\)

Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. \(\mathbf{F}=\langle 0,0,-1\rangle\) across the slanted face of the tetrahedron\(z=4-x-y\) in the first octant; normal vectors point in the positive z-direction.

Describe the surface with the given parametric representation. \(\mathbf{r}(u, v)=\langle u, u+v, 2-u-v\rangle\), for \(0 \leq u \leq 2,0 \leq v \leq 2\)

Find the average value of the function f(x, y, z) = xyz on the unit sphere in the first octant.

Let S be a surface that represents a thin shell with density p. The moments about the coordinate planes (see Section 13.6) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S\), \(M_{x z}=\iint_{S} y \rho(x, y, z) d S, M_{x y}=\iint_{S} z \rho(x, y, z) d S\). The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}\), \(\bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m}\), where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The constant-density cone with radius a, height h, and base in the xy-plane

Give a parametric description for a cylinder with radius a and height h, including the intervals for the parameters.

Give a parametric description for a cone with radius a and height h, including the intervals for the parameters.

Give a parametric description for a sphere with radius a including the intervals for the parameters.

Explain how to compute the surface integral of a scalar-valued function f over a cone using an explicit description of the cone.

Explain how to compute the surface integral of a scalar-valued function f over a sphere using a parametric description of the sphere.

Explain how to compute a flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) over a cone using an explicit description and a given orientation of the cone.

Explain how to compute a surface integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) over a sphere using a parametric description of the sphere and a given orientation.

Explain what it means for a surface to be orientable.

Describe the usual orientation of a closed surface such as a sphere.

Why is the upward flux of a vertical vector field \(\mathbf{F}=\langle 0,0,1\rangle\) across a surface equal to the area of the projection of the surface in the xy-plane?

Give a parametric description of the form \(\mathbf{r}(u, v)=\langle x(u, v), y(u, v), z(u, v)\rangle\) for the following surfaces. The descriptions are not unique. The plane \(2 x-4 y+3 z=16\)

Give a parametric description of the form \(\mathbf{r}(u, v)=\langle x(u, v), y(u, v), z(u, v)\rangle\) for the following surfaces. The descriptions are not unique. The cap of the sphere \(x^{2}+y^{2}+z^{2}=16, \text { for } 4 / \sqrt{2} \leq z \leq 4\)

Give a parametric description of the form \(\mathbf{r}(u, v)=\langle x(u, v), y(u, v), z(u, v)\rangle\) for the following surfaces. The descriptions are not unique. The frustum of the cone \(z^{2}=x^{2}+y^{2}\), for \(2 \leq z \leq 8\)

Give a parametric description of the form \(\mathbf{r}(u, v)=\langle x(u, v), y(u, v), z(u, v)\rangle\) for the following surfaces. The descriptions are not unique. The hyperboloid \(z^{2}=1+x^{2}+y^{2}\), for \(1 \leq z \leq 10\)

Give a parametric description of the form \(\mathbf{r}(u, v)=\langle x(u, v), y(u, v), z(u, v)\rangle\) for the following surfaces. The descriptions are not unique. The portion of the cylinder \(x^{2}+y^{2}=9\) in the first octant, for \(0 \leq z \leq 3\)

Give a parametric description of the form \(\mathbf{r}(u, v)=\langle x(u, v), y(u, v), z(u, v)\rangle\) for the following surfaces. The descriptions are not unique. The cylinder \(y^{2}+z^{2}=36\), for \(0 \leq x \leq 9\)

Find the area of the following surfaces using a parametric description of the surface. The plane \(z=10-x-y\) above the square \(|x| \leq 2,|y| \leq 2\)

Find the area of the following surfaces using a parametric description of the surface. A cone with base radius r and height h, where r and h are positive constants

Find the area of the following surfaces using a parametric description of the surface. The cap of the sphere \(x^{2}+y^{2}+z^{2}=4\), for \(1 \leq z \leq 2\)

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using a parametric description of the surface. \(f(x, y, z)=x^{2}+y^{2}\), where S is the hemisphere \(x^{2}+y^{2}+z^{2}=36\), for \(z \geq 0\)

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using a parametric description of the surface. \(f(x, y, z)=y\), where S is the cylinder \(x^{2}+y^{2}=9,0 \leq z \leq 3\)

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using a parametric description of the surface. \(f(x, y, z)=y\), where S is the cylinder \(x^{2}+z^{2}=1,0 \leq y \leq 3\)

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using a parametric description of the surface. \(f(\rho, \varphi, \theta)=\cos \varphi\), where S is the part of the unit sphere in the first octant

Find the area of the following surfaces using an explicit description of the surface. The cone \(z^{2}=4\left(x^{2}+y^{2}\right)\), for \(0 \leq z \leq 4\)

Find the area of the following surfaces using an explicit description of the surface. The paraboloid \(z=2\left(x^{2}+y^{2}\right)\, for \(0 \leq z \leq 8\)

Find the area of the following surfaces using an explicit description of the surface. The trough \(z=x^{2}\), for \(-2 \leq x \leq 2,0 \leq y \leq 4\)

Find the area of the following surfaces using an explicit description of the surface. The part of the hyperbolic \(z=x^{2}-y^{2}\) above the sector \(R=\{(r, \theta): 0 \leq r \leq 4,-\pi / 4 \leq \theta \leq \pi / 4\}\)

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using an explicit representation of the surface. \(f(x, y, z)=x y\); S is the plane \(z=2-x-y\) in the first octant.

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using an explicit representation of the surface. \(f(x, y, z)=x y\); S is the paraboloid \(z=2-x-y\) for \(0 \leq z \leq 4\)

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using an explicit representation of the surface. \(f(x, y, z)=25-x^{2}-y^{2}\); S is the hemisphere centered at the origin with radius 5, for \(z \geq 0\)

Evaluate the surface integral \(\iint_{S} f(x, y, z) d S\) using an explicit representation of the surface. \(f(x, y, z)=e^{z}\); S is the plane \(z=8-x-2 y\) in the first octant.

Find the average temperature on that part of the plane \(3 x+4 y+z=6\) over the square \(|x| \leq 1,|y| \leq 1\), where the temperature is given by \(T(x, y, z)=e^{-z}\)

Find the average squared distance between the origin and the points on the paraboloid \(z=4-x^{2}-y^{2}\), for \(z \geq 0\).

Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. \(\mathbf{F}=\langle x, y, z\rangle\) across the slanted face of the tetrahedron \(z=10-2 x-5 y\) in the first octant; normal vectors point in the positive z-direction.

Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. \(\mathbf{F}=\langle x, y, z\rangle\) across the slanted surface of the cone \(z^{2}=x^{2}+y^{2}\) for \(0 \leq z \leq 1\); normal vectors point in the positive z-direction.

Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. \(\mathbf{F}=\left\langle e^{-y}, 2 z, x y\right\rangle\) across the curved sides of the surface \(S=\{(x, y, z): z=\cos y,|y| \leq \pi, 0 \leq x \leq 4\}\), where normal vectors point upward.

Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{3}\) across the sphere of radius a centered at the origin, where \(\mathbf{r}=\langle x, y, z)\); normal vectors point outward.

Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. \(\mathbf{F}=\langle-y, x, 1\rangle\) across the cylinder \(y=x^{2}\) for \(0 \leq x \leq 1\), \(0 \leq z \leq 4\); normal vectors point in the positive y-direction.

Determine whether the following statements are true and give an explanation or counterexample. a. If the surface S is given by \(\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1\) \(z=10\}\), then \(\iint_{S} f(x, y, z) d S=\int_{0}^{1} \int_{0}^{1} f(x, y, 10) d x d y .\). b. If the surface S is given by \(\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1\) \(z=x\}\), then \(\iint_{S} f(x, y, z) d S=\int_{0}^{1} \int_{0}^{1} f(x, y, x) d x d y\). c. The surface \(\mathbf{r}=\left\langle v \cos u, v \sin u, v^{2}\right\rangle\), for \(0 \leq u \leq \pi\), \(0 \leq v \leq 2\) is the same as the surface \(\mathbf{r}=\langle\sqrt{v} \cos 2 u\), \(\sqrt{v} \sin 2 u, v)\), for \(0 \leq u \leq \pi / 2,0 \leq v \leq 4\). d. Given the standard parameterization of a sphere, the normal vectors \(\mathbf{t}_{u} \times \mathbf{t}_{v}\) are outward normal vectors.

Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or in the positive z-direction. \(\iint_{S} \nabla \ln r \cdot \mathbf{n} d S\), where S is the hemisphere \(x^{2}+y^{2}+z^{2}=a^{2}\), for \(z \geq 0\), where \(r=|\langle x, y, z\rangle|\)

Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or in the positive z-direction. \(\iint_{S}|\mathbf{r}| d S\), where S is the cylinder \(x^{2}+y^{2}=4\), for \(0 \leq z \leq 8\), where \(\mathbf{r}=\langle x, y, z\rangle\)

Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or in the positive z-direction. \(\iint_{S} x y z d S\), where S is that part of the plane \(z=6-y\) that lies in the cylinder \(x^{2}+y^{2}=4\)

Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or in the positive z-direction. \(\iint_{S} \frac{\langle x, 0, z\rangle}{\sqrt{x^{2}+z^{2}}} \cdot \mathbf{n} d S\), where S is the cylinder \(x^{2}+z^{2}=a^{2}\), \(|y| \leq 2\)

The cone \(z^{2}=x^{2}+y^{2}\), for \(z \geq 0\), cuts the sphere \(x^{2}+y^{2}+z^{2}=16\) along a curve C. a. Find the surface area of the sphere below C, for \(z \geq 0\). b. Find the surface area of the sphere above C. c. Find the surface area of the cone below C, for \(z \geq 0\).

Consider the sphere \(x^{2}+y^{2}+z^{2}=4\) and the cylinder \((x-1)^{2}+y^{2}=1\), for \(z \geq 0\). a. Find the surface area of the cylinder inside the sphere. b. Find the surface area of the sphere inside the cylinder.

Find the upward flux of the field \(\mathbf{F}=\langle x, y, z\rangle\) across the plane \(x / a+y / b+z / c=1\) in the first octant. Show that the flux equal c times the area of the base of the region. Interpret the result physically.

Consider the field \(\mathbf{F}=\langle x, y, z\rangle\) and the cone \(z^{2}=\left(x^{2}+y^{2}\right) / a^{2}\), for \(0 \leq z \leq 1\). a. Show that when \(a=1\), the outward flux across the cone is zero. Interpret the result. b. Find the outward flux (away from the z-axis), for any \(a>0\). Interpret the result.

Find the general formula for the surface area of a cone with height h and base radius a (excluding the base).

A sphere of radius a is sliced parallel to the equatorial plane at a distance a - h from the equatorial plane (see figure). Find the general formula for the surface area of the resulting spherical cap (excluding the base) with thickness h.

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{P}\), where \(\mathbf{r}=\langle x, y, z\rangle\) and p is a real number. Let S be the sphere of radius a centered at the origin. Show that the outward flux of F across the sphere \(4 \pi / a^{p-3}\). It is instructive to do the calculation using both an explicit and parametric description of the sphere.

The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T\), where T (x, y, z) is the temperature in the object and \(k>0\) is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. \(T(x, y, z)=100 e^{-x-y}\); S consists of the faces of the cube \(|x| \leq 1,|y| \leq 1,|z| \leq 1\).

The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T\), where T (x, y, z) is the temperature in the object and \(k>0\) is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. \(T(x, y, z)=100 e^{-x^{2}-y^{2}-z^{2}}\); S is the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\),

The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T\), where T (x, y, z) is the temperature in the object and \(k>0\) is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. \(T(x, y, z)=-\ln \left(x^{2}+y^{2}+z^{2}\right)\); S is the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\).

Let S be the cylinder \(x^{2}+y^{2}=a^{2}\), for \(-L \leq z \leq L\). a. Find the outward flux of the field \(\mathbf{F}=\langle x, y, 0\rangle\) across S. b. Find the outward flux of the field \(\mathbf{F}=\frac{\langle x, y, 0)}{\left(x^{2}+y^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}\) across S, where \(|\mathbf{r}|\) is the distance from the z-axis and p is a real number. c. In part (b), for what values of p is the flux finite as \(a \rightarrow \infty\) (with L fixed)? d. In part (b), for what values of p is the flux finite as \(L \rightarrow \infty\) (with a fixed)?

Let S be a surface that represents a thin shell with density p. The moments about the coordinate planes (see Section 13.6) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S\), \(M_{x z}=\iint_{S} y \rho(x, y, z) d S, M_{x y}=\iint_{S} z \rho(x, y, z) d S\). The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}\), \(\bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m}\), where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The constant-density half cylinder \(x^{2}+z^{2}=a^{2},-h / 2 \leq y \leq h / 2\), \(z \geq 0\)

Let S be a surface that represents a thin shell with density p. The moments about the coordinate planes (see Section 13.6) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S\), \(M_{x z}=\iint_{S} y \rho(x, y, z) d S, M_{x y}=\iint_{S} z \rho(x, y, z) d S\). The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}\), \(\bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m}\), where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The cylinder \(x^{2}+y^{2}=a^{2}, 0 \leq z \leq 2\), with density \(\rho(x, y, z)=1+z\)

Show that \(\left|\mathbf{t}_{u} \times \mathbf{t}_{v}\right|=a^{2} \sin u\) for a sphere of radius a defined parametrically by r(u, v) = \(\langle a \sin u \cos v, a \sin u \sin v, a \cos u\rangle\), where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\).

Suppose that a surface S is defined as \(z=g(x, y)\) on a region R. Show that \(\mathbf{t}_{x} \times \mathbf{t}_{y}=\left(-z_{x},-z_{y}, 1\right)\) and that \(\iint_{S} f(x, y, z) d S=\) \(\iint_{R} f(x, y, z) \sqrt{z_{x}^{2}+z_{y}^{2}+1} d A\).

Let \(y=f(x)\) be a curve in the xy-plane with \(f(x) \neq 0\), for \(a \leq x \leq b\). Let S be the surface generated when the graph of f on [a, b] is revolved about the x-axis. a. Show that S is described parametrically by r(u, v) = \(\langle u, f(u) \cos v, f(u) \sin v\rangle\), for \(a \leq u \leq b, 0 \leq v \leq 2 \pi\). b. Find an integral that gives the surface area of S. c. Apply the result of part (b) to the surface generated with \(f(x)=x^{3}\); for \(1 \leq x \leq 2\). d. Apply the result of part (b) to the surface generated with \(f(x)=\left(25-x^{2}\right)^{1 / 2}\), for \(3 \leq x \leq 4\).

Let z = s(x, y) define a surface over a region R in the xy-plane, where \(z \geq 0\) on R. Show that the downward flux of the vertical vector field \(\mathbf{F}=\langle 0,0,-1\rangle\) across S equals the area of R. Interpret the result physically.

Surface area of a torus a. Show that a torus with radii \(R>r\) (see figure) may be described parametrically by \(r(u, v)=\langle(R+r \cos u) \cos v,(R+r \cos u) \sin v, r \sin u)\), for \(0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi\). b. Show that the surface area of the torus is \(4 \pi^{2} R r\)