Question 1

Find the limit.&#10; &#10;A) Figure 5&#10;B) Figure 8&#10;C) Figure 1&#10;D) Figure 3

Accepted Answer

B

Question 2

Evaluate the line integral along the curve C.&#10;

Accepted Answer

C

Question 3

Find the limit.&#10; &#10;A) Figure 8&#10;B) Figure 3&#10;C) Figure 1&#10;D) Figure 5

Accepted Answer

B

Question 4

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The line integral along the curve $C$ can be evaluated by breaking it into two parts corresponding to $C_1$ and $C_2$, and then summing the results.For $C_1$, parameterize the curve as $r(t) = 6t^2i - 6tj, 0 \leq t \leq 1$. The differential arc length $ds$ can be found by taking the derivative of $r(t)$ with respect to $t$, which gives $r'(t) = 12ti - 6j$, and then calculating its magnitude $|r'(t)| = \sqrt{(12t)^2 + (-6)^2} = \sqrt{144t^2 + 36} = 6\sqrt{t^2 + 1}$. The integral over $C_1$ becomes $\int_0^1 (-6t)(6\sqrt{t^2 + 1}) dt$, since $y = -6t$ and $z = 0$ along $C_1$.For $C_2$, the curve is parameterized as $r(t) = 6i - 6j + (t-1)k, 1 \leq t \leq 2$. Here, $y = -6$ and $z = t - 1$, and the differential arc length $ds$ is simply $dt$ because the path moves only in the $z$-direction. The integral over $C_2$ becomes $\int_1^2 (-6 + t - 1) dt = \int_1^2 (t - 7) dt$.Evaluating these integrals:For $C_1$, the integral is $\int_0^1 -36t\sqrt{t^2 + 1} dt$, which requires a substitution $u = t^2 + 1$, $du = 2t dt$, leading to $-18\int (u^{1/2}) du$, which evaluates to $-18[\frac{2}{3}u^{3/2}]_1^2 = -12[u^{3/2}]_1^2 = -12[(2)^{3/2} - 1^{3/2}]$ = $-12[2\sqrt{2} - 1]$.For $C_2$, the integral is $\int_1^2 (t - 7) dt$, which evaluates directly to $\frac{1}{2}(2^2 - 1^2) - 7(2 - 1) = \frac{1}{2}(3) - 7 = \frac{3}{2} - 7 = -\frac{11}{2}$.The total integral is the sum of the integrals over $C_1$ and $C_2$, which, when calculated correctly, should match one of the given options. The detailed calculation for each part involves integrating the given functions over the specified intervals, which requires some algebraic manipulation and integration techniques. The correct answer involves correctly evaluating these integrals and summing their results. The explanation provided does not directly calculate these integrals to the final numerical answer due to the complexity of the integrals and the algebra involved, but the approach to solving the problem is outlined. The correct choice is based on understanding that the line integral involves integrating the scalar function $y + z$ over the path defined by $C_1$ and $C_2$, and the detailed steps involve specific integration techniques for each segment.

Question 5

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral along the curve...

Question 6

Evaluate the line integral of f(x,y) along the curve C.&#10;

Accepted Answer

We first parameterize the curve C using x as the parameter:
$$x = x$$
$$y = -4x-3$$
$$0 \leq x \leq 3$$

Then, we can calculate the line integral as:
$$\int_C f(x,y) ds = \int_0^3 [(x^2+(-4x-3)^2)\sqrt{1+(-4)^2}] dx$$
$$\int_0^3 [(x^2+16x^2+24x+9)\sqrt{17}] dx$$
$$\int_0^3 [(17x^2+24x+9)\sqrt{17}] dx$$
$$[\frac{17}{3}x^3+12x^2+9x\sqrt{17}]_0^3$$
$$288\sqrt{17}$$

Therefore, the answer is C, 288$\sqrt{17}$.

Question 7

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral along the curve...

Question 8

Evaluate the line integral along the curve C.&#10;

Accepted Answer

Using the formula for line integrals, we have:

$$\int_C \left(\frac{4}{z}\right)^{1/3} ds = \int_a^b \left(\frac{4}{z(t)}\right)^{1/3} |\mathbf{r}'(t)| dt$$

where $z(t) = 4t^3$, $\mathbf{r}'(t) = (12t^2\cos t)\mathbf{i} + (12t^2\sin t)\mathbf{j} + (12t^2)\mathbf{k}$, and $|\mathbf{r}'(t)| = 12t^2$.

Plugging these into the integral, we get:

$$\int_0^{3\sqrt{2}} \left(\frac{4}{4t^3}\right)^{1/3} (12t^2) dt = \int_0^{3\sqrt{2}} 4\ dt = 12\sqrt{2}$$

Therefore, the correct answer is D) $72(4-\sqrt{2})$.

Question 9

Find the limit.&#10; &#10;A) Figure 7&#10;B) Figure 4&#10;C) Figure 1&#10;D) Figure 2

Accepted Answer

The vector function $ r(t) = \sin t \, \mathbf{j} - \cos t \, \mathbf{k} $ traces a quarter-circle in the $ jk $-plane from $ (0, -1) $ to $ (1, 0) $ as $ t $ goes from 0 to $\frac{\pi}{2}$. This corresponds to Figure 2.

Question 10

Find the limit.&#10; &#10;A) Figure 2&#10;B) Figure 8&#10;C) Figure 4&#10;D) Figure 7

Accepted Answer

The path traced by $\mathbf{r}(\mathrm{t})$ is a helix with axis along the $\mathbf{j}$ direction. As $\mathrm{t}$ approaches $\pm \infty$, $\mathbf{r}(\mathrm{t})$ spirals along the helix and does not approach any finite point. However, as $\mathrm{t}$ approaches $\pm 1$, $\mathbf{r}(\mathrm{t})$ approaches the point $(0,\pm 3,0)$ on the $\mathbf{j}$ axis. Looking at the answer choices, only Figure 7 depicts a helix with axis along the $\mathbf{j}$ direction and approaches the points $(0,3,0)$ and $(0,-3,0)$ as $\mathrm{t}$ approaches $\pm 1$. Thus, the answer is D.

Question 11

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral along the curve...

Question 12

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral along the curve...

Question 13

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral along the curve...

Question 14

Find the limit.&#10; &#10;A) Figure 8&#10;B) Figure 1&#10;C) Figure 3&#10;D) Figure 5

Accepted Answer

The answer of Find the limit.&#10; &#10;A) Figure 8&#10;B) Figure...

Question 15

Find the limit.&#10; &#10;A) Figure 2&#10;B) Figure 6&#10;C) Figure 4&#10;D) Figure 7

Accepted Answer

The answer of Find the limit.&#10; &#10;A) Figure 2&#10;B) Figure...

Question 16

Evaluate the line integral of f(x,y) along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral of f(x,y) along...

Question 17

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral along the curve...

Question 18

Find the limit.&#10; &#10;A) Figure 3&#10;B) Figure 6&#10;C) Figure 5&#10;D) Figure 1

Accepted Answer

The answer of Find the limit.&#10; &#10;A) Figure 3&#10;B) Figure...

Question 19

Find the limit.&#10; &#10;A) Figure 5&#10;B) Figure 3&#10;C) Figure 8&#10;D) Figure 6

Accepted Answer

The answer of Find the limit.&#10; &#10;A) Figure 5&#10;B) Figure...

Question 20

Evaluate the line integral along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral along the curve...

Question 21

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Accepted Answer

The answer of 11eb49f8_e04b_996a_b71a_d56c76eb7903_TB6591_0011eb49f8_e04b_c07c_b71a_c5c95e2f64bc_TB6591_00...

Question 22

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Accepted Answer

The answer of 11eb49f8_e04b_996a_b71a_d56c76eb7903_TB6591_0011eb49f8_e04b_c07d_b71a_a9e22c6d6691_TB6591_00...

Question 23

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Accepted Answer

The answer of 11eb49f8_e04b_996a_b71a_d56c76eb7903_TB6591_0011eb49f8_e04b_c07b_b71a_a3ed57b729fc_TB6591_00...

Question 24

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Accepted Answer

The answer of 11eb49f8_e04b_9965_b71a_c17f2d5463c4_TB6591_0011eb49f8_e04b_9967_b71a_af686d52705c_TB6591_00...

Question 25

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Accepted Answer

The answer of 11eb49f8_e04b_c07f_b71a_b5a1df82cc5b_TB6591_0011eb49f8_e04b_e793_b71a_e1466b150328_TB6591_00...

Question 26

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Accepted Answer

The answer of 11eb49f8_e04b_c07f_b71a_b5a1df82cc5b_TB6591_0011eb49f8_e04b_e791_b71a_811c6e7c6fb2_TB6591_00...

Question 27

Evaluate the line integral of f(x,y) along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral of f(x,y) along...

Question 28

Evaluate the line integral of f(x,y) along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral of f(x,y) along...

Question 29

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Accepted Answer

The answer of 11eb49f8_e04b_c07f_b71a_b5a1df82cc5b_TB6591_0011eb49f8_e04b_e792_b71a_d1a997d83851_TB6591_00...

Question 30

Evaluate the line integral of f(x,y) along the curve C.&#10; &#10;A) 200&#10;B) 400&#10;C) 0&#10;D) 100

Accepted Answer

The answer of Evaluate the line integral of f(x,y) along...

Question 31

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Accepted Answer

The answer of 11eb49f8_e04b_9965_b71a_c17f2d5463c4_TB6591_0011eb49f8_e04b_9968_b71a_1fa1b99c8d02_TB6591_00...

Question 32

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Accepted Answer

The answer of 11eb49f8_e04b_9965_b71a_c17f2d5463c4_TB6591_0011eb49f8_e04b_9969_b71a_4bf4f62497b5_TB6591_00...

Question 33

Evaluate the line integral of f(x,y) along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral of f(x,y) along...

Question 34

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Accepted Answer

The answer of 11eb49f8_e04b_c07f_b71a_b5a1df82cc5b_TB6591_0011eb49f8_e04b_e790_b71a_8b965023fb2a_TB6591_00...

Question 35

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Accepted Answer

The answer of 11eb49f8_e04b_9965_b71a_c17f2d5463c4_TB6591_0011eb49f8_e04b_9966_b71a_b1dd190b3b4f_TB6591_00...

Question 36

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 37

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Accepted Answer

The answer of 11eb49f8_e04b_996a_b71a_d56c76eb7903_TB6591_0011eb49f8_e04b_c07e_b71a_5b6c19ddea09_TB6591_00...

Question 38

Evaluate the line integral of f(x,y) along the curve C.&#10;

Accepted Answer

The answer of Evaluate the line integral of f(x,y) along...

Question 39

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 40

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 41

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 42

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 43

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 44

Calculate the circulation of the field F around the closed curve C.&#10;

Accepted Answer

The answer of Calculate the circulation of the field F...

Question 45

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 46

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 47

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 48

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 49

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 50

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 51

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 52

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 53

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 54

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 55

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 56

Find the work done by F over the curve in the direction of increasing t.&#10;

Accepted Answer

The answer of Find the work done by F over...

Question 57

Calculate the circulation of the field F around the closed curve C.&#10;

Accepted Answer

The answer of Calculate the circulation of the field F...

Question 58

Find the gradient field of the function.&#10;f(x, y, z) = z sin (x + y + z)

Accepted Answer

The answer of Find the gradient field of the function.&#10;f(x,...

Question 59

Calculate the circulation of the field F around the closed curve C.&#10;

Accepted Answer

The answer of Calculate the circulation of the field F...

Question 60

Find the gradient field of the function.&#10;

Accepted Answer

The answer of Find the gradient field of the function.&#10;...

Question 61

Calculate the flow in the field F along the path C.&#10;

Accepted Answer

The answer of Calculate the flow in the field F...

Question 62

Calculate the flow in the field F along the path C.&#10; &#10;A) 5&#10;B) -4&#10;C) - 7&#10;D) 28

Accepted Answer

The answer of Calculate the flow in the field F...

Question 63

Calculate the flow in the field F along the path C.&#10;

Accepted Answer

The answer of Calculate the flow in the field F...

Question 64

Calculate the flux of the field F across the closed plane curve C.&#10; &#10;A) 0&#10;B) -288&#10;C) 144&#10;D) -144

Accepted Answer

The answer of Calculate the flux of the field F...

Deck 17: Integrals and Vector Fields