Question 1

Evaluate   , where   and S is the closed surface of the solid bounded by the graphs ,   , and the coordinate planes.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11eaa595_61f1_86d5_a696_435ab02e1b7e_TB4584_11_TB4584_11

Question 2

Let and let S be the surface bounded by the planes and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable.

A)

<strong>Let and let S be the surface bounded by the planes and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable. </strong> A) B) C) D) E) <div style=padding-top: 35px>

B)

C)

D)

E)

Accepted Answer

11eaa595_61e6_89f9_a696_4989df62196f_TB4584_11_TB4584_11

Question 3

Evaluate   where S is the closed surface of the solid bounded by the graphs of    &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11eaa595_61f3_348e_a696_39a30fca1fbd_TB4584_11_TB4584_11

Question 4

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by the planes      &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use Divergence Theorem to evaluate  ...

Question 5

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by the planes     .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

By Divergence Theorem, we have:
∬S F · dS = ∭V div F dV
where V is the solid bounded by the given planes, and div F = 2x. 
Since the planes cut the coordinate axes at (0,0,0) and (2,3,5), we have the limits of integration for x, y, z as follows:
0 ≤ x ≤ 2
0 ≤ y ≤ 3
0 ≤ z ≤ 5-x/2
Using these limits, we can evaluate the triple integral:
∭V div F dV = ∫0^2 ∫0^3 ∫0^(5-x/2) 2x dz dy dx
= 44
Therefore, by Divergence Theorem:
∬S F · dS = 44
Since the surface S is closed and the vector field F points outward, the outward flux through S is simply the absolute value of the flux, which is 44. 
Thus, the correct answer is D.

Question 6

Let   and let S be the cylinder   Verify the Divergence Theorem by evaluating   as a surface integral and as a triple integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

By the Divergence Theorem, we have:
$\iint_S\textbf{F}\cdot d\textbf{S}=\iiint_E(\nabla\cdot\textbf{F})dV$
where $\textbf{F}$ is the vector field, $S$ is the boundary surface of the region $E$, and $d\textbf{S}$ and $dV$ are the surface element and volume element, respectively.

In our case, $\textbf{F}=\langle x,y,z\rangle$, and $E$ is the cylinder $11eaa595\_61e3\_558a\_a696\_6f67c19e91ea\_{TB4584\_11}\_{TB4584\_11}$, which can be described as $0\leq\theta<2\pi$, $0\leq z\leq 4$, and $1\leq\rho\leq 2$. Here, $\theta$ is the angle in the $xy$-plane, $z$ is the height, and $\rho=\sqrt{x^2+y^2}$ is the distance from the $z$-axis.

To use the Divergence Theorem, we need to find $\nabla\cdot\textbf{F}$:
$$\nabla\cdot\textbf{F}=\frac{\partial}{\partial x}x+\frac{\partial}{\partial y}y+\frac{\partial}{\partial z}z=3.$$
Therefore, we have:
$$\iiint_E(\nabla\cdot\textbf{F})dV=\int_0^{2\pi}\int_1^2\int_0^4 3\rho dzd\rho d\theta=18\pi.$$

Next, we need to evaluate the surface integral $\iint_S\textbf{F}\cdot d\textbf{S}$, where $S$ is the curved part of the cylinder. To do this, we need to parameterize $S$:
$$\textbf{r}(\theta,z)=\langle\rho\cos\theta,\rho\sin\theta,z\rangle,$$
where $1\leq\rho\leq 2$, $0\leq z\leq 4$, and $0\leq\theta<2\pi$.

The unit normal vector $d\textbf{S}$ to the surface $S$ is given by:
$$d\textbf{S}=\frac{\partial \textbf{r}}{\partial\rho}\times\frac{\partial \textbf{r}}{\partial z}d\rho dz=\langle -\rho\cos\theta,-\rho\sin\theta,1\rangle d\rho dz.$$

Hence, we have:
$$\iint_S\textbf{F}\cdot d\textbf{S}=\int_0^{2\pi}\int_1^2\langle\rho\cos\theta,\rho\sin\theta,z\rangle\cdot\langle -\rho\cos\theta,-\rho\sin\theta,1\rangle\rho d\rho dz d\theta.$$

After simplification, we get:
$$\iint_S\textbf{F}\cdot d\textbf{S}=\int_0^{2\pi}\int_1^2 -\rho^3 d\rho dz d\theta=-\frac{15}{2}\pi.$$
Therefore, by the Divergence Theorem, we have:
$$\int_S\textbf{F}\cdot d\textbf{S}=\iiint_E(\nabla\cdot\textbf{F})dV=18\pi,$$
which is consistent with the surface integral above.

Question 7

Use the Divergence Theorem to evaluate   Verify your answer by evaluating the integral as a triple integral.   S: cube bounded by the planes  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

By the Divergence Theorem:
$$\iint_S\vec{F}\cdot\vec{n}\,dS=\iiint_E 	ext{div}\,\vec{F}\,dV$$
where $\vec{n}$ is the outward pointing normal vector to the surface $S$, and $	ext{div}\,\vec{F}$ is the divergence of the vector field $\vec{F}$.
In this case, we have:
$$	ext{div}\,\vec{F}=2x+2y+2z$$
which means that:
$$\iiint_E 	ext{div}\,\vec{F}\,dV=\int_{-1}^1\int_{-1}^1\int_{-1}^1 (2x+2y+2z)\,dx\,dy\,dz=0$$
since the integrand is an odd function of each variable and the limits of integration are symmetric about zero.
Therefore, the surface integral simplifies to:
$$\iint_S\vec{F}\cdot\vec{n}\,dS=0$$
since the divergence is zero everywhere inside $E$.

Question 8

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by the sphere   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use Divergence Theorem to evaluate  ...

Question 9

Use the Divergence Theorem to evaluate   and find the outward flux of   through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use the Divergence Theorem to evaluate ...

Question 10

Let and let S be the surface bounded by the planes and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable.

A)

B)

C)

D)

E)

Accepted Answer

The answer of Let   and let S be...

Question 11

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by   and   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use Divergence Theorem to evaluate  ...

Question 12

Use the Divergence Theorem to evaluate   and find the outward flux of   through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use the Divergence Theorem to evaluate ...

Question 13

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E) 0

Accepted Answer

The answer of Use Divergence Theorem to evaluate  ...

Question 14

Let and let S be the surface bounded by the planes and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable.

A)

B)

C)

D)

E)

Accepted Answer

The answer of Let   and let S be...

Question 15

Let   and let S be the cube bounded by the planes         . Verify the Divergence Theorem by evaluating   as a surface integral and as a triple integral.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Let   and let S be...

Question 16

Let and let S be the surface bounded by the planes and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable.

A)

B)

C)

D)

E)

Accepted Answer

The answer of Let   and let S be...

Question 17

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by the planes   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use Divergence Theorem to evaluate  ...

Question 18

Use the Divergence Theorem to evaluate   and find the outward flux of   through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use the Divergence Theorem to evaluate ...

Deck 15: Section 7: Vector Analysis