Question 3 B0/1 pto 10 99 Details Consider the vector field F = (xy, **y) Is this vector field Conservative? Select an answer v If so: Find a function f so that F = vf + K f(x,y) = Use your answer t

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Answer 1

The vector field F = (x*y, y) is not conservative.

To determine if the vector field F = (x*y, y) is conservative, we can check if its curl is zero. The curl of a 2D vector field F = (P(x, y), Q(x, y)) is given by:
Curl(F) = (∂Q/∂x) - (∂P/∂y)

In our case, P(x, y) = x*y and Q(x, y) = y. So we need to compute the partial derivatives:
∂P/∂y = x
∂Q/∂x = 0

Now, we can compute the curl:
Curl(F) = (∂Q/∂x) - (∂P/∂y) = 0 - x = -x

Since the curl is not zero, we can state that the vector field F is not conservative.

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Related Questions

Find the exact value of each expression a. cos(105) b. sin(%) and Find the exact value of each of the angles that should be written using radian measure a. sin" (-0,5) b. cos(0)

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the exact values are:

a. cos(105) = (√2 - √6)/4

b. The exact value of sin(%) depends on the specific value of the angle %.

c. sin^(-1)(-0.5) = -pi/6 radians

d. cos(0) = 1.

To find the exact value of cos(105), we can use the cosine addition formula:

Cos(A + B) = cos(A)cos(B) – sin(A)sin(B)

In this case, we can write 105 as the sum of 60 and 45 degrees:

Cos(105) = cos(60 + 45)

Using the cosine addition formula:

Cos(105) = cos(60)cos(45) – sin(60)sin(45)

We know the exact values of cos(60) and sin(45) from special right triangles:

Cos(60) = ½

Sin(45) = √2/2

Substituting these values:

Cos(105) = (1/2)(√2/2) – (√3/2)(√2/2)

= √2/4 - √6/4

= (√2 - √6)/4

b. To find the exact value of sin(%), we need to know the specific value of the angle %. Without that information, we cannot determine the exact value.

c. For the angle in radians, we have:

a. sin^(-1)(-0.5)

The value sin^(-1)(-0.5) represents the angle whose sine is -0.5. From the unit circle or trigonometric identity, we know that sin(pi/6) = ½. Since sine is an odd function, sin(-pi/6) = -1/2. Therefore, sin^(-1)(-0.5) = -pi/6 radians.

c. Cos(0)

The value cos(0) represents the cosine of the angle 0 radians. From the unit circle or trigonometric identity, we know that cos(0) = 1.

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Prove that the sequence {an} with an = sin(nt/2) is divergent. ( =

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The sequence [tex]$\{a_n\}$[/tex] with [tex]$a_n = \sin\left(\frac{nt}{2}\right)$[/tex] is divergent.

What is the divergence of a sequence?

The divergence of a sequence refers to a situation where the terms of the sequence do not approach a specific limit as the index of the sequence increases indefinitely. In other words, if a sequence does not converge to a finite value or approach positive or negative infinity, it is considered divergent.

To prove that the sequence [tex]$\{a_n\}$[/tex] with [tex]$a_n = \sin\left(\frac{nt}{2}\right)$[/tex] is divergent, we can show that it does not converge to a specific limit.

Suppose [tex]$\{a_n\}$[/tex] is a convergent sequence with limit [tex]$L$.[/tex] Then for any positive value [tex]$\varepsilon > 0$[/tex], there exists a positive integer [tex]$N$[/tex]such that for all[tex]$n > N$, $|a_n - L| < \varepsilon$.[/tex]

Let's choose[tex]$\varepsilon = 1$[/tex]for simplicity. Now, we need to find an integer[tex]$N$[/tex] such that for all [tex]$n > N$, $|a_n - L| < 1$.[/tex]

Consider the term[tex]$a_{2N}$[/tex] in the sequence. We have:

[tex]\[a_{2N} = \sin\left(\frac{2Nt}{2}\right) = \sin(Nt)\][/tex]

Since the sine function is periodic with a period of [tex]$2\pi$[/tex], the values of [tex]$\sin(Nt)$[/tex] will repeat for different values of [tex]$N$[/tex] and [tex]$t$.[/tex]

Let [tex]$t = \frac{\pi}{2N}$[/tex]. Then we have:

[tex]\[a_{2N} = \sin\left(\frac{N\pi}{2N}\right) = \sin\left(\frac{\pi}{2}\right) = 1\][/tex]

So, we can choose [tex]$N$[/tex] such that [tex]$2N > N$[/tex]and[tex]$|a_{2N} - L| = |1 - L| < 1$.[/tex]

However, for[tex]$a_{2N + 1}$,[/tex] we have:

[tex]\[a_{2N + 1} = \sin\left(\frac{(2N + 1)t}{2}\right) = \sin\left(\frac{(2N + 1)\pi}{4N}\right)\][/tex]

The values of [tex]$\sin\left(\frac{(2N + 1)\pi}{4N}\right)$[/tex] will vary as $N$ increases. In particular, as $N$ becomes very large,[tex]$\sin\left(\frac{(2N + 1)\pi}{4N}\right)$[/tex]oscillates between -1 and 1, never converging to a specific value.

Thus, we have shown that for any chosen limit $L$, there exists an[tex]$\varepsilon = 1$[/tex] such that there is no $N$ satisfying[tex]$|a_n - L| < 1$ for all $n > N$.[/tex]

Therefore, the sequence [tex]$\{a_n\}$[/tex] with [tex]$a_n = \sin\left(\frac{nt}{2}\right)$[/tex] is divergent.

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12. [0/5 Points] DETAILS PREVIOUS ANSWERS UD 82 n The series Σ is e3n n=1 O divergent by the Comparison Test divergent by the Test for Divergence a convergent geometric series divergent by the Integr

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The series Σ e^3n/n, n=1, is divergent by the Test for Divergence. the Test for Divergence states that if the limit of the terms of a series does not approach zero, then the series is divergent. In this case, as n approaches infinity, the term e^3n/n does not approach zero. Therefore, the series is divergent.

The series Σ e^3n/n, n=1, is divergent because the terms of the series do not approach zero as n approaches infinity. The Test for Divergence states that if the limit of the terms does not approach zero, the series is divergent. In this case, the term e^3n/n does not approach zero because the exponential growth of e^3n overwhelms the linear growth of n. Consequently, the series does not converge to a finite value and is considered divergent.

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Consider the following limits of exponential functions and
compute the limit with explanation, thank you!
Consider the following limits of exponential functions. You are welcome to use a graph or a numerical chart for intution, but be sure to justify your reasoning algebraically. (a) limei (Hint: recall p

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The characteristics of exponential functions can be used to evaluate the limit (lim_xtoinfty ex).

The exponential function (ex) rises without limit as x approaches infinity. This may be seen by looking at the graph of "(ex)," which demonstrates that the function quickly increases as "(x)" becomes greater.

We may defend this mathematically by taking into account the exponential function's definition. A quantity's exponential development is represented by the value of (ex), where (e) is the natural logarithm's base. Exponent x increases as x grows larger, and the function ex grows exponentially as x rises in size.

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Question 5 Not yet answered The graph of y = /(x) passes through the points (1.5) and (3, 11). The tangent line to y = f(x) at (3, 11) has the equation: y = -x + 7. a) What is the average rate of change of f(x) on the interval 1 SXS 3? b) What is the instantaneous rate of change of f(x) at the point (3, 11)? Explain c) Explain why f(x) has a critical number in the interval 1 s * $ 3. You can assume that f'(X) is continuous. In your explanation use the The Mean Value Theorem, to argue that for some c. S'C) = 3. Then use the Intermediate Value Theorem applied to f'(x) to argue that for some d. /'(d) = 0 Points out of 3.00 Flag question Maximum file size: 500MB, maximum number of files: 1 Files You can drag and drop Niles here to add them. Accepted file types PDF document pat Question 6 Not yet answered Points out of 200 Find an equation of the tangent line to the graph of x - y - 26 ot(3, 1). Show your work for full credit Maximum file size: 600MB, maximum number of files: 1 Files Pro question You can drag and drop files hore to add them. Accepted file types PDF documentadt

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a) The average f(x) change rate across the range [1, 3] is 2.

To find the average rate of change of f(x) on the interval [1, 3], we use the formula:

Average rate of change = (f(3) - f(1))/(3 - 1)

Given that f(3) = 11 and f(1) = 7 (from the equation of the tangent line), we can substitute these values into the formula:

Average rate of change = (11 - 7)/(3 - 1) = 4/2 = 2

Therefore, the average rate of change of f(x) on the interval [1, 3] is 2.

b) The instantaneous rate of change of f(x) at the point (3, 11) is -1 because the tangent line's slope is -1.

The instantaneous rate of change of f(x) at the point (3, 11) can be found by taking the derivative of the function f(x) and evaluating it at x = 3.

However, since the equation of the tangent line y = -x + 7 is already given, we can directly determine the slope of the tangent line, which represents the instantaneous rate of change at that point.

The slope of the tangent line is -1, so the instantaneous rate of change of f(x) at the point (3, 11) is -1.

c) We want to show that f(x) has a critical number in the interval [1, 3]. According to the Mean Value Theorem, if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the interval (a, b) such that the instantaneous rate of change at c is equal to the average rate of change over the interval [a, b].

In this case, we have already determined that the average rate of change of f(x) on the interval [1, 3] is 2. Since the instantaneous rate of change of f(x) at x = 3 is -1, and the function f(x) is continuous on the interval [1, 3], by the Mean Value Theorem, there exists at least one point c in the interval (1, 3) such that the instantaneous rate of change at c is equal to 2.

Now, let's consider the function f'(x), which represents the instantaneous rate of change of f(x) at each point. Since f'(3) = -1 and f'(1) = 2, the function f'(x) is continuous on the closed interval [1, 3] (as it is the tangent line to f(x) at each point).

According to the Intermediate Value Theorem, if a function f(x) is continuous on the closed interval [a, b], and k is any number between f(a) and f(b), then there exists at least one point d in the interval (a, b) such that f'(d) = k.

In this case, since -1 is between f'(1) = 2 and f'(3) = -1, the Intermediate Value Theorem guarantees the existence of a point d in the interval (1, 3) such that f'(d) = -1. Therefore, f(x) has a critical number in the interval [1, 3].

Note: The question also mentions using the Mean Value Theorem to argue for the existence of a point c such that f'(c) = 3. However, this is incorrect as the given equation of the tangent line y = -x + 7 does not have a slope of 3.

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The box plot displays the cost of a movie ticket in several cities.

A box plot uses a number line from 3 to 25 with tick marks every one unit. The box extends from 8 to 14 on the number line. A line in the box is at 10. The lines outside the box end at 6 and 22. The graph is titled Movie Ticket Prices, and the line is labeled Cost Of Tickets.

Which of the following is the best measure of center for the data shown, and what is that value?

The median is the best measure of center and equals 10.
The median is the best measure of center and equals 11.
The mean is the best measure of center and equals 10.
The mean is the best measure of center and equals 11.

Answers

The median is the best measure of center for the given data and the value of the median is 10.

solve for the vertex of f(x)=x^2-10x+13 using completing the square

Answers

Answer:

(5, 38)

Step-by-step explanation:

To find the vertices of the quadratic function f(x) = x^2 - 10x + 13 using squared interpolation, do the following:

step 1:

Group the terms x^2 and x.

f(x) = (x^2 - 10x) + 13

Step 2:

Complete the rectangle for the grouped terms. To do this, take half the coefficients of the x term, square them, and add them to both sides of the equation.

f(x) = (x^2 - 10x + (-10/2)^2) + 13 + (-10/2)^2

= (x^2 - 10x + 25) + 13 + 25

Step 3:

Simplify the equation.

f(x) = (x - 5)^2 + 38

Step 4:

The vertex form of the quadratic function is f(x) = a(x - h)^2 + k. where (h,k) represents the vertex of the parabola. Comparing this to the simplified equation shows that the function vertex is f(x) = x^2 - 10x + 13 (h, k) = (5, 38).

So the vertex of the quadratic function is (5, 38).

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. �
2
4
+
�
2
49
=
1
4
x 2

+ 49
y 2

=1

Answers

In summary:

- The major axis has end points (-2, 0) and (2, 0).

- The minor axis has end points (0, -7) and (0, 7).

- This ellipse does not have real foci.

The equation of the ellipse in standard form is:

(x^2/4) + (y^2/49) = 1

In this form, the major axis is along the x-axis, and the minor axis is along the y-axis.

To identify the end points of the major and minor axes, we need to find the values of a and b, which are the lengths of the semi-major and semi-minor axes, respectively.

For this ellipse, a = 2 and b = 7 (square root of 49).

Therefore, the end points of the major axis are (-2, 0) and (2, 0), and the end points of the minor axis are (0, -7) and (0, 7).

To find the foci of the ellipse, we can calculate c using the formula:

c = sqrt(a^2 - b^2)

In this case, c = sqrt(4 - 49) = sqrt(-45).

Since the value under the square root is negative, it means that this ellipse does not have real foci.

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Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). a) Find the area of the face of the section superimposed on the rectangular coordinate system. b) Find the weight of the section Round your answer to three decimal places. ya 2+ 2 m -6 -4-2 2 6. (-5.5, 0) 4 (5.5, 0)

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To find the area of the face of the section superimposed on the rectangular coordinate system, we need to break down the shape into smaller rectangles and triangles and calculate their individual areas.

To find the weight of the section, we need to know the material density and thickness of the section. Multiplying the density by the volume of the section will give us the weight. The volume can be calculated by finding the sum of the individual volumes of the smaller rectangles and triangles within the section.

a) To find the area of the face of the section, we can break it down into smaller rectangles and triangles. We calculate the area of each shape individually and then sum them up. In the given figure, we can see rectangles and triangles on both sides of the y-axis. By calculating the areas of these shapes, we can find the total area of the section superimposed on the rectangular coordinate system.

b) To find the weight of the section, we need additional information such as the density and thickness of the material. Once we have this information, we can calculate the volume of each individual shape within the section by multiplying the area by the thickness. Then, we sum up the volumes of all the shapes to obtain the total volume. Finally, multiplying the density by the total volume will give us the weight of the section.

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In fact, take a guess at the limit_lim (√²+2-√√²+2) and then evaluate it to see if you guessed correctly

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Lets take a guess at the the limit of the expression √²+2-√√²+2 to be 1.

To evaluate the limit of the given expression, we can substitute a value for the variable that approaches the limit.

Let's consider x as the variable. As x approaches 0, the expression becomes √(x^2+2) - √(√(x^2+2)).

To simplify the expression, we can use the property √a - √b = (√a - √b)(√a + √b)/(√a + √b). Applying this property, we get (√(x^2+2) - √(√(x^2+2))) = [(√(x^2+2) - √(√(x^2+2))) * (√(x^2+2) + √(√(x^2+2))))/((√(x^2+2) + √(√(x^2+2)))).

By simplifying further, we obtain (x^2 + 2 - √(x^2+2))/(√(x^2+2) + √(√(x^2+2))).

Taking the limit as x approaches 0, we substitute 0 for x in the expression, resulting in (0^2 + 2 - √(0^2+2))/(√(0^2+2) + √(√(0^2+2))). This simplifies to (2 - 2)/(√2 + √2) = 0/2 = 0.

Therefore, the limit of √²+2-√√²+2 as x approaches 0 is 0.

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4.
Use natural logarithms to solve the equation. Round to the nearest thousandth.

2e2x + 5 = 10

−1.695

1.007

0.402

0.458

Answers

The natural logarithm of the both sides of the exponential function indicates that the value of x in the equation is the option;

0.458

What is an exponential function?

An exponential function is a function of the form f(x) = eˣ, where x is the value of the input variable.

The exponential equation can be presented as follows;

[tex]2\cdot e^{2\cdot x}[/tex] + 5 = 10

The value of x can be found using natural logarithm as follows;

[tex]2\cdot e^{2\cdot x}[/tex] = 10 - 5 = 5

[tex]e^{2\cdot x}[/tex] = 5/2 = 2.5

ln([tex]e^{2\cdot x}[/tex]) = ln(2.5)

2·x = ln(2.5)

x = ln(2.5)/2 ≈ 0.458

The value of x in the equation [tex]2\cdot e^{2\cdot x}[/tex] + 5 = 10 is; x = 0.458

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What happens to the value of the digits in a number when the number is divided by 10^1?
A.
Each digit has a value that is 1/1,000 of its value in the original number.
B.
Each digit has a value that is 10 times its value in the original number.
C.
Each digit has a value that is 1/10 of its value in the original number.
D.
Each digit has a value that is 1/100 of its value in the original number.

Answers

When a number is divided by [tex]10^1[/tex] (10), each digit in the number has a value that is 1/10 of its value in the original number. Thus, the correct answer is option C: Each digit has a value that is 1/10 of its value in the original number.

When a number is divided by [tex]10^1[/tex] (which is 10), the value of each digit in the number is reduced by a factor of 10.

To understand this, let's consider a number with digits in the place value system. Each digit represents a specific value based on its position in the number. For example, in the number 1234, the digit '1' represents 1000, the digit '2' represents 200, the digit '3' represents 30, and the digit '4' represents 4.

When we divide this number by 10^1 (which is 10), we are essentially shifting all the digits one place to the right. In other words, we are moving the decimal point one place to the left. The result would be 123.4.

Now, let's observe the changes in the digit values:

The digit '1' in the original number had a value of 1000, and in the result, it has a value of 10. So, its value has decreased by a factor of 10 (1/10).

The digit '2' in the original number had a value of 200, and in the result, it has a value of 2. So, its value has also decreased by a factor of 10 (1/10).

The digit '3' in the original number had a value of 30, and in the result, it has a value of 0.3. So, its value has also decreased by a factor of 10 (1/10).

The digit '4' in the original number had a value of 4, and in the result, it has a value of 0.04. So, its value has also decreased by a factor of 10 (1/10).

Therefore, when a number is divided by [tex]10^1[/tex] (10), each digit in the number has a value that is 1/10 of its value in the original number. Thus, the correct answer is option C: Each digit has a value that is 1/10 of its value in the original number.

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Prove the remaining part of theorem 4.2.4: if f:A->B with Rng(f)=C, and if f^-1is a function, then f○f^-1=I[C].

Answers

The remaining part of Theorem 4.2.4 states that if f: A -> B is a function with range C and its inverse function f^(-1) exists, then the composition of f with f^(-1) is equal to the identity function on the range C, denoted as I[C].

To prove this, let's consider the composition f○f^(-1). By the definition of function composition, for any c in C, we need to show that (f○f^(-1))(c) = IC, where I[C] is the identity function on C.

Since f is a function with range C, every element in C has a preimage in A. Let's take an arbitrary element c in C. Since f^(-1) is a function, we can apply it to c to obtain f^(-1)(c), which lies in A. Now, applying f to f^(-1)(c), we get f(f^(-1)(c)). Since f^(-1)(c) is in the domain of f, the composition is well-defined.

By the definition of the inverse function, f(f^(-1)(c)) = c for all c in C. This means that (f○f^(-1))(c) = c, which is precisely the definition of the identity function on C, denoted as I[C].

Hence, we have shown that for any c in C, (f○f^(-1))(c) = IC, which implies that f○f^(-1) = I[C]. Thus, we have proven the remaining part of Theorem 4.2.4.

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1- Find the derivative of the following functions: f(x) = x3 + 2x2 +1, f(x) = log(4x + 3), f(x) = sin(x2 + 2), f(x) = 5 In(x-3) 2- Evaluate the following integrals: § 4 ln(x) dx, S(X6 – 2x) dat 2 3

Answers

The integrals of A is 4 * (x * ln(x) - x) + C and The integrals of B is (1/7) * x⁷ - (1/2) * x⁴ + C.

1. Finding the derivatives:

a. f(x) = x³ + 2x² + 1

f'(x) = 3x² + 4x

b. f(x) = log(4x + 3)

f'(x) = 4 / (4x + 3)

c. f(x) = sin(x² + 2)

f'(x) = cos(x² + 2) * 2x

d. f(x) = 5 * ln(x-3)²

To find the derivative of this function, we can apply the chain rule:

Let u = ln(x-3)², then f(x) = 5 * u

Applying the chain rule:

f'(x) = 5 * (du/dx)

= 5 * (2 * ln(x-3) * (1/(x-3)))

= 10 * ln(x-3) / (x-3)

2. Evaluating the integrals:

a. ∫4 ln(x) dx

This integral can be evaluated using integration by parts:

Let u = ln(x) and dv = dx

Then, du = (1/x) dx and v = x

Applying the integration by parts formula:

∫ u dv = uv - ∫ v du

∫4 ln(x) dx = 4 * (x * ln(x) - ∫ x * (1/x) dx)

= 4 * (x * ln(x) - ∫ dx)

= 4 * (x * ln(x) - x) + C

b. ∫(x⁶ - 2x³) dx

To integrate this polynomial, we can use the power rule for integration:

∫ xⁿ dx = (x^(n+1))/(n+1) + C

Applying the power rule:

∫(x⁶ - 2x³) dx = (x⁷)/7 - (2x⁴)/4 + C

= (1/7) * x⁷ - (1/2) * x⁴ + C

Please note that C represents the constant of integration.

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Evaluate SSS 4xy dv where E is the region bounded by z = 2x2 + 2y2 - 7 and z = 1. O a. O O b. -32 3 Oc 128 3 Od. 64 64

Answers

To evaluate the triple integral of 4xy over the region E bounded by z = [tex]2x^2 + 2y^2 - 7[/tex] and z = 1, we need to set up the integral in terms of the appropriate limits of integration.

First, let's consider the limits for the x, y, and z variables:

For z, the lower limit is z = 1 and the upper limit is given by the equation of the upper surface, which is [tex]z = 2x^2 + 2y^2 - 7.[/tex]

For y, the limits are determined by the region E projected onto the yz-plane. To find these limits, we set z = 1 in the equation of the upper surface and solve for y:

[tex]2x^2 + 2y^2 - 7 = 12y^2 = 6 - 2x^2y^2 = 3 - x^2y = ±sqrt(3 - x^2[/tex])

Since the region E is symmetric with respect to the y-axis, we only need to consider the positive values of y.

For x, the limits are determined by the region E projected onto the xz-plane. To find these limits, we set y = 0 in the equation of the upper surface and solve for x:

[tex]2x^2 + 2(0)^2 - 7 = 12x^2 - 6 = 12x^2 = 7x^2 = 7/2x = ±sqrt(7/2)[/tex]

Again, since the region E is symmetric with respect to the x-axis, we only need to consider the positive values of x.

Now we can set up the triple integral:

[tex]∭E 4xy dv = ∫∫∫E 4xy dz dy dx[/tex]

Using the limits we derived earlier, the integral becomes:

[tex]∫(x=sqrt(7/2) to x=0) ∫(y=0 to y=sqrt(3-x^2)) ∫(z=1 to z=2x^2 + 2y^2 - 7) 4xy dz dy dx[/tex]

To evaluate this integral, you would need to perform the integration step by step. The final answer will be one of the options provided (a, b, c, or d).

Please note that without specific numerical values for the options, I cannot directly determine the correct answer for you. You would need to evaluate the integral and compare the result with the given options to determine the correct answer.