1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Luden [163]
3 years ago
10

Let R be the region in the first quadrant of the​ xy-plane bounded by the hyperbolas xyequals​1, xyequals9​, and the lines yequa

ls​x, yequals4x. Use the transformation x equals StartFraction u Over v EndFraction ​, y equals uv with ugreater than0 and vgreater than0 to rewrite the integral below over an appropriate region G in the​ uv-plane. Then evaluate the​ uv-integral over G.
Mathematics
1 answer:
Tema [17]3 years ago
4 0

Answer:

The area can be written as

\int\limits_1^2 \int\limits_1^3 u(\frac{1}{v} - v \, ln(v)) \, du \, dv = 0.2274

And the value of it is approximately 1.8117

Step-by-step explanation:

x = u/v

y = uv

Lets analyze the lines bordering R replacing x and y by their respective expressions with u and v.

  • x*y = u/v * uv = u², therefore, x*y = 1 when u² = 1. Also x*y = 9 if and only if u² = 9
  • x=y only if u/v = uv, And that only holds if u = 0 or 1/v = v, and 1/v = v if and only if v² = 1. Similarly y = 4x if and only if 4u/v = uv if and only if v² = 4

Therefore, u² should range between 1 and 9 and v² ranges between 1 and 4. This means that u is between 1 and 3 and v is between 1 and 2 (we are not taking negative values).

Lets compute the partial derivates of x and y over u and v

x_u = 1/v

x_v = u*ln(v)

y_u = v

y_v = u

Therefore, the Jacobian matrix is

\left[\begin{array}{ccc}\frac{1}{v}&u \, ln(v)\\v&u\end{array}\right]

and its determinant is u/v - uv * ln(v) = u * (1/v - v ln(v))

In order to compute the integral, we can find primitives for u and (1/v-v ln(v)) (which can be separated in 1/v and -vln(v) ). For u it is u²/2. For 1/v it is ln(v), and for -vln(v) , we can solve it by using integration by parts:

\int -v \, ln(v) \, dv = - (\frac{v^2 \, ln(v)}{2} - \int \frac{v^2}{2v} \, dv) = \frac{v^2}{4} - \frac{v^2 \, ln(v)}{2}

Therefore,

\int\limits_1^2 \int\limits_1^3 u(\frac{1}{v} - v \, ln(v)) \, du \, dv = \int\limits_1^2 (\frac{1}{v} - v \, ln(v) ) (\frac{u^2}{2}\, |_{u=1}^{u=3}) \, dv= \\4* \int\limits_1^2 (\frac{1}{v} - v\,ln(v)) \, dv = 4*(ln(v) + \frac{v^2}{4} - \frac{v^2\,ln(v)}{2} \, |_{v=1}^{v=2}) = 0.2274

You might be interested in
1. Look for a pattern in the first three equations. Then, fill in the missing numbers in the rest of the
Alla [95]

Answer:

1=1 times 2/2 1 + 2=2 times 3/2 1 +2+3=3 times 4/2 1+2+3+4=4 times 5/2 1+2+3+4+5=5 times 6/2

4 0
3 years ago
-
Lelechka [254]

B is the correct answer. 3(x+1)=24

6 0
3 years ago
There are 9 female performers in a dance recital. The ratio of men to women is 2:3. How many men are in the dance recital?
Firdavs [7]

Answer:

6 men

Step-by-step explanation:

4 0
3 years ago
If y varies directly as x^2 find k when x=2 and y=8
Grace [21]
I think the answer is 2
8 0
3 years ago
What is the value of 7x 2squared + 2x when x = 3?
mash [69]

Answer:

Given expression has the value 69

Step-by-step explanation:

Given equation is:

=7x^2 + 2x

Now we have to put x = 3

So the equation will become:

=7(3)^2 + 2(3)

By simplifying:

As 3^{2} = 9

and 2*3 = 6

So the above equation will become:

=7(9 ) + 6\\= 63 + 6\\= 69

So the value of given expression is 69.

i hope it will help you!

6 0
3 years ago
Other questions:
  • Multiply and simplify. 6xy^3z⋅3x^2yx^2
    15·1 answer
  • One more than three times a number is the same as the number decreased by 5 find the number​
    9·2 answers
  • A number n divided by 4
    8·2 answers
  • Can i please get help with this question?
    13·2 answers
  • An equation allows you to find the x- and y-coordinates of any point on the xy-plane.
    14·2 answers
  • Can someone help me with this math? explain your answer too.​
    14·1 answer
  • Help ASAP i have a few mins left
    10·2 answers
  • A satellite can move at 16,000 miles per hour. How many hours will it take to reach a star that is two light years away?
    7·1 answer
  • If my dot plot is not skewed or symmetrical which measure of variability should I use? (15 data points) please help asap! :)
    5·1 answer
  • If f(c)=2x^2-x, find f(-4) - f(9)
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!