MATHEMATICS COLLEGE

Compute the area of the region D bounded by xy=1, xy=16, xy2=1, xy2=36 in the first quadrant of the xy-plane. Using the non-linear change of variables u=xy and v=xy2, find x and y as functions of u and v.x=x(u,v)= ?

y=y(u,v)=?

Find the determinant of the Jacobian for this change of variables.

∣∣∣∂(x,y)/∂(u,v)∣∣∣=det=?

Using the change of variables, set up a double integral for calculating the area of the region D.

∫∫Ddxdy=?

Evaluate the double integral and compute the area of the region D.

Area =

Answers

Answer 1

Answer:

Answer:

53.7528

Step-by-step explanation:

Notice that when

$xy = 1 ,\,\,\, xy = 16 , \,\,\, xy^2 = 1 \,\,\,, xy^2 = 36 \n\n$

If you set

$u = xy , v = xy^2$

as they suggest, then

$y = (v)/(u)} \,\,\,\, \text{and} \,\,\,\, \n\n{\displaystyle x = (u)/(y) = (u)/(v/u) = (u^2)/(v)$

Then

${\diplaystyle (\partial(x,y))/(\partial(u,v))} =\det \begin{pmatrix} 2u/v && -u^2/v^2 \n -v/u^2 && 1/u \end{pmatrix} = (1)/(v) }$

Therefore

$\iint\limits_(D) dx\,dy = \int\limits_(1)^(36)\int\limits_(1)^(16) (1)/(v) \, du \, dv = 15 \ln(36) = 53.7528$

Answer 2

Answer:

A Jacobian matrix is formed by the first partial derivatives of a multivariate function that utilizes a training algorithm, and further calculation as follows:

Jacobian:

To evaluate the integral, cover the bounds, the integrand, and the differential area dA.

Transform the four equations in terms of u and v, notice that $u= xy \ \ and \ \ xy = 1, xy = 16$

implies that $1\leq u \leq 16.$

Similarly, $v= xy^2\ \ and\ \ xy^2= 1 , xy^2= 25$ implies that $1 \leq v \leq 25$

so write this integration region as $S= {(u,v) |1 \leq u \leq 18, 1 \leq v \leq 25}.$

Translate the equations from uv - plane to xy- plane. It is obtained by solving,

$u= xy, y= xy^2 \n\n\left.\begin{matrix}u=xy & \n v=xy^2& \end{matrix}\right\} \to \left.\begin{matrix}u^2=x^2y^2 & \n v=xy^2& \end{matrix}\right\} \n\n\to x=(u^2)/(v), y=(v)/(u)$

Convert dA part of the integral , using is $dA= |(\partial (x,y))/(\partial(u,v))| dudv.$

That is, $dA= \begin{vmatrix}(\partial x)/(\partial u) & (\partial x)/(\partial v)\n (\partial y)/(\partial u) & (\partial y)/(\partial v) \end{vmatrix} \ du dv \n\n$

Sampule the partial derivatives to find the Jacobian.

$dA=\begin{vmatrix}(2u)/(v) &-(u^2)/(v^2) \n -(v)/(u^2) &(1)/(u) \end{vmatrix} \ dudv\n\n=[((2u)/(v)) ((1)/(u)) -(- (u^2)/(v^2))(-(v)/(u^2))]\ du dv\n\n=((2)/(v)- (1)/(v)) \ dudv\n\n=(1)/(v)\ du dv\n\n$

The Jacobian the transformation is $dA= (1)/(v)dudv$

The region is $S={(u,v) |1\leq u \leq 16, 1\leq v\leq 25}.$

Rewrite the integral, using the transformation: $S,\ x=(u^2)/(v) =, y=(v)/(u) \ \ and\ \ dA=(1)/(v) dudv\n\n\int\int_R 1dA =\int \int_S (1)/(v)\ dudv= \int^(25)_(1) \int^(16)_(1) \ (1)/(v) \ dudv\n\n$

Evaluate the inner integral with respect to u.

$\to \int\int_R 1dA = \int^(25)_(1) \int^(16)_(1) \ (1)/(v) \ dudv\n\n$

by solving the value we get

$= 30 \ ln (5) \approx 48.28$

Find out more about the Jacobians here:

brainly.com/question/9381576

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Evaluate the expression for m=4;-48/m

Answers

Answer:

-12

Step-by-step explanation:

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Which of the following ratios is not equivalent to 1.22?

Answers

Answer:

Where are options

Step-by-step explanation:

You can edit your question or resend it

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Answers

To solve the problem shown in the figure above you must keep on mind the following information:

1. The figure shows a parabola whose vertex is (0,0).

2. The x² indicates that the red parabola shown in the figure is vertical

3. and the sign - indicates that the red parabola opens down.

Therefore, you can conclude that the red parabola has the equation f(x)=-x²

So, the answer is B.

If we observe the graph of F(x) and G(x), F(x) can be obtained by shifting the graph of G(x) 4 units down.

Shifting 4 units down means subtracting 4 from the function value.

So, G(x) = 4 - x²

Thus,

F(x) = G(x) - 4
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Therefore, the correct answer is option B

Two adjacent angles form a right angle. The larger angle is 3 times the measure of the smaller angle. Which is the measure of the larger angle?This means the angles are in a ratio of 1:3.

A. 67.5°
B. 30°
C. 22.5°
D. 45°

Answers

The angles is B: 30 degrees

Answer:

B. 30

Step-by-step explanation:

When we multiply a number by 3, wesometimes/always/never v

get the same value as if we added 6

to that number.

Stuck? Watch a video or use a hint.

Report a problem

7 of 7 ..

nyone, anywhere

Imnact

Math by grace

O

Answers

Answer:

? what's the question??????????????????

Final answer:

Multiplying a number by 3 usually does not yield the same result as adding 6 to it, except in the case of the number 3. For all other numbers, the results are different.

Explanation:

In mathematics, when we multiply a number by 3 it does not usually yield the same value as when we add 6 to the number. However, there is one instance in which this statement is incorrect. Let's consider the number 3. If we multiply 3 by 3 we get 9, and if we add 6 to 3 we also get 9. In all other instances, multiplying a number by 3 will not yield the same result as adding 6 to that number. For example, if we take the number 4, multiplying it by 3 gives us 12, but adding 6 to it gives us 10, which are different.

Learn more about Multiplication here:

brainly.com/question/35502092

#SPJ11

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