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
Kazeer [188]
3 years ago
5

Use the given transformation to evaluate the given integral, where r is the triangular region with vertices (0, 0), (8, 1), and

(1, 8). (x − 3y) da r , x = 8u + v, y = u + 8v
Mathematics
1 answer:
Jlenok [28]3 years ago
6 0
We first obtain the equation of the lines bounding R.

For the line with points (0, 0) and (8, 1), the equation is given by:

\frac{y}{x} = \frac{1}{8}  \\  \\ \Rightarrow x=8y \\  \\ \Rightarrow8u+v=8(u+8v)=8u+64v \\  \\ \Rightarrow v=0

For the line with points (0, 0) and (1, 8), the equation is given by:

\frac{y}{x} = \frac{8}{1}  \\  \\ \Rightarrow y=8x \\  \\ \Rightarrow u+8v=8(8u+v)=64u+8v \\  \\ \Rightarrow u=0

For the line with points (8, 1) and (1, 8), the equation is given by:

\frac{y-1}{x-8} = \frac{8-1}{1-8} = \frac{7}{-7} =-1 \\  \\ \Rightarrow y-1=-x+8 \\  \\ \Rightarrow y=-x+9 \\  \\ \Rightarrow u+8v=-8u-v+9 \\  \\ \Rightarrow u=1-v

The Jacobian determinant is given by

\left|\begin{array}{cc} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\end{array}\right| = \left|\begin{array}{cc} 8 &1\\1&8\end{array}\right| \\  \\ =64-1=63

The integrand x - 3y is transformed as 8u + v - 3(u + 8v) = 8u + v - 3u - 24v = 5u - 23v

Therefore, the integration is given by:

63 \int\limits^1_0 \int\limits^{1}_0 {(5u-23v)} \, dudv =63 \int\limits^1_0\left[\frac{5}{2}u^2-23uv\right]^{1}_0 \\  \\ =63\int\limits^1_0(\frac{5}{2}-23v)dv=63\left[\frac{5}{2}v-\frac{23}{2}v^2\right]^1_0=63\left(\frac{5}{2}-\frac{23}{2}\right) \\  \\ =63(-9)=|-576|=576
You might be interested in
The charts show the effect of an interest rate adjustment on an adjustable-rate mortgage.
Tcecarenko [31]

Answer:

C. a higher monthly payment.

Step-by-step explanation:

<h3>Answer options:</h3>

Rate adjustment resulted in

<u>A. a lower interest rate. </u>

  • No, interest rate increased from 0.4 to 0.5 percent

<u>B. a higher principal. </u>

  • No, principal decreased from $200000 to $187000

<u>C. a higher monthly payment. </u>

  • Yes, monthly payment increased from $1059.85 to $1162.56

<u>D. a higher number of payments.</u>

  • No, number of payments remained as 336
3 0
3 years ago
HELP!!!!
aniked [119]

Answer:

A. Plane B because it was 9.33 miles away

B. 48 units

Step-by-step explanation:

A. Since the airplanes fly at an angle to the runway, their direction forms a triangle with the runway with their height above the ground as the opposite of the angle and their distance from the airport as the hypotenuse.

So for airplane A with 44° angle of departure,

sin44° = y/h where y = height above the ground and h = distance from airport

So h = y/sin44° = 6/sin44° = 8.64 miles

So for airplane B with 40° angle of departure,

sin40° = y/H where y = height above the ground and H = distance from airport

So H = y/sin40° = 6/sin40° = 9.33 miles

Since airplane B is at 9.33 miles away from the airport whereas airplane A is 8.64 miles from the airport, airplane B is farther away.

B. We know that scale factor = new size/original size

Our scale factor = 4 and original size = 12 units. So,

new size = scale factor original size = 4 × 12 = 48 units.

8 0
2 years ago
A rectangle has a length of 6cm by 8cm. Which is the length of a diagonal of the rectangle?
balandron [24]
A,10 cm if you like this answer explained to you let me know
7 0
3 years ago
Which inequality represents this sentence ? 7 5 is greater than 6 2
Blizzard [7]
75 > 62
the way I was taught it is that the < and > are alligators and the alligator mouth always opens for the bigger number, because he is hungry
6 0
2 years ago
Pls help <br> What is side AC
Maslowich

Answer:

tan●=opposite/adjacent

tan70=AC/3

AC=3tan70

AC=8.24

4 0
2 years ago
Other questions:
  • Guests had to choose their meals from beef, chicken or vegetarian.
    8·1 answer
  • Which undefined geometric term is described as a location on a coordinated plane that designated by an ordered pair,(x,y)?
    7·2 answers
  • Round 521,742 to the nearest thousand.
    6·2 answers
  • I'll GIVE BRAINLIST TO THE FRIST PERSON!!!!!!!!!!!!!
    13·2 answers
  • If 588 digits were used to number the pages of a book, how many pages are in this book?
    8·2 answers
  • Alyssa is planning to have a lemonade stand. Alyssa and her mom make 165 ounces of lemonade. If Alyssa fills each cup equally wi
    6·1 answer
  • At Brick House Masonry Supply, it is time for annual pay raises for the fork lift operators.
    6·1 answer
  • An item that originally costs $12.00 was on the clearance aisle for $3.60. What is the percentage of discount?
    6·1 answer
  • Helpppp<br> Solve for x.
    5·1 answer
  • Look at the image for any information and the question thank you
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!