Ignore the top part. <br> but can someone help me with part b please?

Use the given transformation to evaluate the integral. double integral 9xy dA R , where R is the region in the first quadrant bo

lianna [129]

It looks like the boundaries of $R$ are the lines $y=\dfrac23x$ and $y=3x$ , as well as the hyperbolas $xy=\frac23$ and $xy=3$ . Naturally, the domain of integration is the set

$R = \left\{(x,y) ~:~ \dfrac{2x}3 \le y \le 3x \text{ and } \dfrac23 \le xy \le 3 \right\}$

By substituting $x=\frac uv$ and $y=v$ , so $xy=u$ , we have

$\dfrac23 \le xy \le 3 \implies \dfrac23 \le u \le 3$

and

$\dfrac{2x}3 \le y \le 3x \implies \dfrac{2u}{3v} \le v \le \dfrac{3u}v \implies \dfrac{2u}3 \le v^2 \le 3u \implies \sqrt{\dfrac{2u}3} \le v \le \sqrt{3u}$

so that

$R = \left\{(u,v) ~:~ \dfrac23 \le u \le 3 \text{ and } \sqrt{\dfrac{2u}3 \le v \le \sqrt{3u}\right\}$

Compute the Jacobian for this transformation and its determinant.

$J = \begin{bmatrix}x_u & x_v \\ y_u & y_v\end{bmatrix} = \begin{bmatrix}\dfrac1v & -\dfrac u{v^2} \\\\ 0 & 1 \end{bmatrix} \implies \det(J) = \dfrac1v$

Then the area element under this change of variables is

$dA = dx\,dy = \dfrac{du\,dv}v$

and the integral transforms to

$\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \int_{\sqrt{2u/3}}^{\sqrt{3u}} \frac{dv\,du}v$

Now compute it.

$\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \ln|v|\bigg|_{v=\sqrt{2u/3}}^{v=\sqrt{3u}} \,du \\\\ ~~~~~~~~ = \int_{2/3}^3 \ln\left(\sqrt{3u}\right) - \ln\left(\sqrt{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln(3u) - \ln\left(\frac{2u}3\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln\left(\frac{3u}{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \int_{2/3}^3 du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \left(3-\frac23\right) = \boxed{\frac76 \ln\left(\frac92\right)}$

7 0

2 years ago

It is required to make 14 grams of certain chemical compound called Z. This is made from compounds W, X, and Y in the ratio of 2

dexar [7]

Answer:

W = 9.33 grams

X = 4.67 grams

Step-by-step explanation:

Given;

Mass of Z to be made = 14 grams

Composition of Z

W:X:Y = 2:1:3

To make 6 grams of Y requires 4 grams of W and 2 grams of X

Composition of Y (for 6grams)

W:X = 4:2 = 2:1

Substituting the composition of Y to Z

Z = W:X:Y = 2:1:3

Z = W:X:W:X = 2:1:2:1

summing them up, we have;

Z = W:X = 4:2 = 2:1

Since the final mixing ratio for W and X are 2:1, to make 14 grams of Z, we need;

W = 2/3 × 14grams = 28/3 grams

W = 9.33 grams

X = 1/3 × 14grams = 14/3 grams

X = 4.67 grams

8 0

3 years ago

Kareem is participating in a 5-day cross-country biking challenge. He biked for 67, 62, 59, and 47 miles on the first four days.

BabaBlast [244]

Answer:65miles

Step-by-step explanation:

Let the needed miles be x

(67+62+59+47+x)/5=60

235+x=60×5

235+x=300

x=300-235

x=65miles

6 0

3 years ago

Find y. Question attatched.

ICE Princess25 [194]

Answer: y=4

Step-by-step explanation:

We have a khown angle wich is 30 degree .

Cos 30 = root square of 3 /2

So root sqaure of 3/2 = 4 root square 3 / x

X= 4 r.sq 3 / r.sq 3/2 = 8

So we just need to apply the pythagorian theorem

X^2= y^2+ 48

64 = y^2 +48

Y^2 = 16

Y=4

4 0

3 years ago

Order -3.98, 3 8/9, -3 11/12, and 3.9 repeated from least to greatest

Mkey [24]

Answer:

Step-by-step explanation:

-3.98

3 8/9 = 3.888 (repeating)

-3

3.9

least to greatest :

-3.98 , -3, 3 8/9, 3.9

3 0

3 years ago

Ignore the top part. but can someone help me with part b please?