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
AnnyKZ [126]
3 years ago
5

Calculate s f(x, y, z) ds for the given surface and function. g(r, θ) = (r cos θ, r sin θ, θ), 0 ≤ r ≤ 4, 0 ≤ θ ≤ 2π; f(x, y, z)

= x2 + y2
Mathematics
1 answer:
Triss [41]3 years ago
4 0

g(r,\theta)=(r\cos\theta,r\sin\theta,\theta)\implies\begin{cases}g_r=(\cos\theta,\sin\theta,0)\\g_\theta=(-r\sin\theta,r\cos\theta,1)\end{cases}

The surface element is

\mathrm dS=\|g_r\times g_\theta\|\,\mathrm dr\,\mathrm d\theta=\sqrt{1+r^2}\,\mathrm dr\,\mathrm d\theta

and the integral is

\displaystyle\iint_Sx^2+y^2\,\mathrm dS=\int_0^{2\pi}\int_0^4((r\cos\theta)^2+(r\sin\theta)^2)\sqrt{1+r^2}\,\mathrm dr\,\mathrm d\theta

=\displaystyle2\pi\int_0^4r^2\sqrt{1+r^2}\,\mathrm dr=\frac\pi4(132\sqrt{17}-\sinh^{-1}4)

###

To compute the last integral, you can integrate by parts:

u=r\implies\mathrm du=\mathrm dr

\mathrm dv=r\sqrt{1+r^2}\,\mathrm dr\implies v=\dfrac13(1+r^2)^{3/2}

\displaystyle\int_0^4r^2\sqrt{1+r^2}\,\mathrm dr=\frac r3(1+r^2)^{3/2}\bigg|_0^4-\frac13\int_0^4(1+r^2)^{3/2}\,\mathrm dr

For this integral, consider a substitution of

r=\sinh s\implies\mathrm dr=\cosh s\,\mathrm ds

\displaystyle\int_0^4(1+r^2)^{3/2}\,\mathrm dr=\int_0^{\sinh^{-1}4}(1+\sinh^2s)^{3/2}\cosh s\,\mathrm ds

\displaystyle=\int_0^{\sinh^{-1}4}\cosh^4s\,\mathrm ds

=\displaystyle\frac18\int_0^{\sinh^{-1}4}(3+4\cosh2s+\cosh4s)\,\mathrm ds

and the result above follows.

You might be interested in
BEST ANSWER GETS BRAINLIEST!! PLEASE HELP
ddd [48]
(1/3) * (3^3) + (5.2) * 2 = 19.4
5 0
3 years ago
The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall sem
Sophie [7]

Answer:

Variance =10900.00

Standard deviation=104.50

Step by step Explanation:

Admissions Probability for 1100= 0.2

Admissions Probability for 1400=0.3

Admissions Probability for 1300 =0.5

To find the expected value, we will multiply each possibility by its probability and then add.

mean = 1100*0.2 + 1400*0.3 + 1300*0.5 = 1290

To find the variance, we will start by squaring each possibility and then multiplying it by its probability. We will then add these and subtract the mean squared.

E(X^2)=( 1100²*0.2)+ (1400²*0.3 )+ (1300²*0.5) = 1675000

Variance(X)=E(X²)- [E(X)]²

= 1675000 - (1290)²

=10900

Hence, the Variance(X)=10900

Then to calculate the standard variation , we will use the formular below,

standard variation (X)=√ var(X)= √10900

=104.5

Hence the standard variation=104.5

8 0
3 years ago
Based on the table of values below, find the slope between points where x = 5 and
aleksklad [387]

Answer:

Slope = 1

Step-by-step explanation:

The table gives the values of y corresponding to the values of x.

At x =5, y = 2.

At x = 7, y = 6 and  

At x = 10, y = 7

Now, we have to determine the slope between points where x = 5 and where x = 10.

Now, the two concerned points are (5,2) and (10,7) and we have to find the slope between those two points.

Now, slope,  m = \frac{7 - 2}{10 - 5} = 1 (Answer)

3 0
3 years ago
Read 2 more answers
Multiply 6.75x10^6 and 1.2x10^6 Keep the numbers written in scientific notation when you are solving the problem Show all of you
harkovskaia [24]

Answer:

8.1×10^12

Step-by-step explanation:

6.75×1.2=8.1 and power should be added 6+6=12 and answer is 8.1×10^12

4 0
2 years ago
<img src="https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B36%7D%7B169%7D%7D" id="TexFormula1" title="\sqrt{\frac{36}{169}}" alt="\sq
otez555 [7]

Answer:

0.0355029585799?

Step-by-step explanation:

5 0
2 years ago
Other questions:
  • Look at the table. Make a conjecture about the sum of the first 25 positive even numbers.
    7·2 answers
  • Leala took five tests in science. Find her average score to the nearest tenth:
    9·2 answers
  • Which system is represented by the graph?
    10·1 answer
  • An input - output table for algebric
    10·2 answers
  • I need help ASAP please
    15·1 answer
  • For the sets: A = {1, 2, 3, 4, 5} B = {1, 3, 5} C = {4, 6} U = {numbers from 0 to 10},
    11·1 answer
  • The measure of the exterior angle of the triangle is what?
    11·1 answer
  • Write an equation in standard form.
    12·2 answers
  • What is the value of y?<br> Thankss
    11·2 answers
  • A group of friends wants to go to the amusement park. They have $207.50 to spend on parking and admission. Parking is $5, and ti
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!