Compute the integral Z Z Z U y dV , where U is the part of the ball of radius 2, centered at (0, 0, 0), that lies in the 1st octant. Recall that the first octant is the part of the 3d space where all three coordinates x, y, z are nonnegative. (Hint: You may use cylindrical or spherical coordinates for this computation, but note that the computation with cylindrical coordinates will involve a trigonometric substitution - so spherical cooridnates should be preferable.)

Answers

Answer 1

As the hint suggests, convert to spherical coordinates using

x = p cos(u) sin(v)

y = p sin(u) sin(v)

z = p cos(v)

dV = dx dy dz = p² sin(v) dp du dv

Then U is the set

[tex]U = \left\{ (p,u,v) \mid 0\le p\le2 \text{ and } 0\le u\le \dfrac{\pi}2 \text{ and } 0\le v\le\dfrac{\pi}2\right\}[/tex]

and the integral of y over U is

[tex]\displaystyle \iiint_U y \, dV = \iiint_U p\sin(u)\sin(v) \cdot p^2 \sin(v) \, dV[/tex]

[tex]\displaystyle \iiint_U y \, dV = \int_0^{\frac\pi2} \int_0^{\frac\pi2} \int_0^2 p^3 \sin(u) \sin^2(v) \, dp \, du \, dv [/tex]

[tex] \displaystyle \iiint_U y \, dV = \frac{2^4-0^4}4 \int_0^{\frac\pi2} \int_0^{\frac\pi2} \sin(u) \sin^2(v) \, du \, dv [/tex]

[tex]\displaystyle \iiint_U y \, dV = 4 \cdot \left(-\cos\left(\frac\pi2\right) + \cos(0)\right) \int_0^{\frac\pi2} \sin^2(v) \, dv [/tex]

[tex]\displaystyle \iiint_U y \, dV = 4 \cdot \frac12 \int_0^{\frac\pi2} (1-\cos(2v)) \, dv [/tex]

[tex]\displaystyle \iiint_U y \, dV = 2 \left(\left(\frac\pi2 - \frac12 \sin\left(2\cdot\frac\pi2\right)\right) - \left(0 - \frac12 \sin\left(2\cdot0\right)\right) \right)[/tex]

[tex]\displaystyle \iiint_U y \, dV = \pi - \sin(\pi) = \boxed{\pi}[/tex]


Related Questions

What function is graphed below?

Answers

its the third option

Use a protractor to measure the
angle. Then identify it as either
acute, right, obtuse, or straight.
Name
Size

Answers

Answer:

Straight angle 180 degrees

Kenny has 8 red shirts, 12 blue shirts, and 10 white shirts in his closet. What is the probability that he would get a white shirt out of the closet?

Answers

Answer:

1/3 or 33.33 %

Step-by-step explanation:

this is because there is 30 shirts and all and 10 white. 10/30 is 1/3, or 33.33%

Answer:

likely

If there are 8 red shirts, 12 blue shirts, and 10 white shirts. The answer would be likely

Anna adds 2oz of milk to an 18oz cup of coffee that is 10% milk. What is the percentage of the milk in the resulting mixture?

Answers

Answer:

19% of milk

lemme know if it helped

Step-by-step explanation:

For one English test jason correctly answered 22 questions these correct answers gave him a percent score of 50% In total how many questions were on this English test

Answers

There had to of been 40 questions or 50

(5 1/4−2 1/2)+(6 2/3−3 1/2) plsss help (real answers)

Answers

Answer:

5 11/12. hope this helps

Step-by-step explanation:

Answer:

6

Step-by-step explanation: 5 1/4 - 2/12 = 3    6 2/3 - 3 1/2=3  3+3=6

Hello everybody! Can anybody help me out with this fairly simple multiple choice algebra problem? Thanks! ​

Answers

Answer:

l = P -2w

________ D is the correct choice

2

find the value of the angles marked 'x' and 'y' ​

Answers

Answer:

x=120 y=60

Step-by-step explanation:

Interior angle of a hexagon = 120

Straight line = 180

180-120=60

Help please I’ll give points help help help help

Answers

Answer:

my sample ka

Step-by-step explanation:

bigyan mu ako para masagot ko ang tanong nayan.

Original Price of a truck $17,100.00 Tax:2.4%

Answers

Final Price: 17,510.40 dollars

You multiple the tax by the original to get the amount of tax. Then you add it to your original to get total price.

Answer:

17504.40

Step-by-step explanation:

You take 17100 x 2.4% and get 410.40

add 17100 and 410.40 and get 17504.40

If f(x)=2(x2+5), then f(-3)=

Answers

Answer:

-1

Step-by-step explanation:

So you would do

-3×2= -6

then do

-6+5= -1

3
1) Wendy has 3/5 gallon of milk. She
used 1/8
gallon to make cookies. How
much milk does Wendy have left?
8
a)
T-
WIN
b)
b) io
3
1
19
29
c)
d)
40
40

Answers

Answer: 19/40 gallons

Step-by-step explanation:

She has 3/5. She used 1/8. 3/5=24/40 and 1/8 = 5/40.

24/40 - 5/40 = 19/40 gallons

:0 hope it helps

Each week, Christian earns $x fixing bicycles and receives $10 as his allowance.
To find how much he will earn in total over 2 weeks, Christian calculates the following:
x + 10 + x + 10
What is another way Christian can calculate how much he will earn in total over the 2 weeks?
Drag and drop the equivalent expression in the box.

Answers

Answer:

2(x+10)

Step-by-step explanation:

x + 10 + x + 10 is the same as 2 times the expression x + 10

This can be written as 2(x+10)

-Chetan K

Answer: 2(x+10)

Step-by-step explanation:

x + 10 + x + 10 is the same as 2 times the expression x + 10

This can be written as 2(x+10)

50 point question Solve quick i will give branliest

Answers

Answer:

17

Step-by-step explanation:

x + 100 = 4x + 49

100 - 49 = 3x

3x = 51

x = 17

jessie works in an electronics store and earns a 7.5% commission on every item that she sells. How much commission would she earn for selling a computer that costs $1,580?

Answers

Answer: $118.50

Work Shown:

7.5% = (7.5)/100 = 0.075

7.5% of $1580 = 0.075*1580 = 118.50

What's 9 + 10?


Whoever gets it right get's the crown!

Answers

Answer:

jokes aside it is 19

Step-by-step explanation:

but if it is jokes then 21

Answer:

Step-by-step explanation:

the answer would be 21

Amana Hagel earns $476 a week. She is married and claims 3 allowances. How much will
be withheld from her weekly paycheck for Federal Income Tax?

Answers

Answer:

$158.67

Step-by-step explanation:

Hi , do you love math ?

Answers

It’s not my favorite subject. Sometimes it could be easy but other times it could be really difficult for me.

Answer: yes math is so good

Step-by-step explanation:

HELP ME OUT PLEASE!!

For the formula shown, how many sulfur atoms are represented?

O 1

O 2

O 4

O 6​

Answers

The answer would be 1 because sulfur is “S” and as you can see there is no number beside S meaning there is only 1 sulfur while there is 2 hydrogen and 4 oxygen

H₂SO₄ is Sulfuric Acid and the number of atoms Sulfur has is 4 atoms

What is a Chemical Formula?

A chemical formula is a formula that lists the constituent parts of a compound together with their relative proportions. No subscript is used if there is just one atom of a certain kind. A subscript is added to the symbol for an atom if it contains two or more of a certain type of atom.

Given data ,

The chemical is represented as H₂SO₄  which is Sulfuric Acid

Sulfuric acid has 2 atoms of Hydrogen , 1 atom of Sulfur and 4 atoms of Oxygen

As per the chemical formula , we can see that there is no subscript added to the Sulfur S in the chemical formula equation.

Therefore , The number of Sulfur atoms present in H₂SO₄  is 1

Hence , H₂SO₄  has only 1 sulfur atom

To learn more about chemical formula click :

https://brainly.com/question/29031056

#SPJ2

pls help (picture) *30 points*

Answers

Answer:

perimeter = 36

Step-by-step explanation:

Each diagonal side is the hypotenuse of a right triangle with legs of lengths 6 and 8.

c² = a² + b²

c² = 6² + 8²

c² = 36 + 64

c² = 100

c = √100

c = 10

Each horizontal side measures 8.

perimeter = 8 + 8 + 10 + 10

perimeter = 36

June earned $ 75 last week. April earned $25. June earnings were how many times greater than April

Answers

Answer:

3 times

Step-by-step explanation:

25 times 3 is 75

This is what i think

The radius of a circle is 16 ft. Find its area in terms of \pi

Answers

256 pi ft

For this problem, first we need to recognize that the area of a circle is found using the equation: pi r^2

When we plug in the value of r:
pi 16^2

We get an answer of 256 pi ft

Ellen solved the equation x2−5=59 using the following steps.

Review her work carefully, and then follow the instructions.

Line 1: x2−5=59
Line 2: x2−5+5=59+5
Line 3: x2=64
Line 4: x=8
Which statement identifies a mistake Ellen made, if any?

Answers

Answer:

c

Step-by-step explanation:

In Line 4, she forgot to take the negative square root, because squaring both 8 and −8 equals 64.

Find the value of X Show all work
(2x+6)
36
(4x-18)

Answers

Answer:

X=30

Step-by-step explanation:

(2X + 6) + 36 = (4X- 18)

36=4X-18-2X-6

36=2X-24

2X = 60

X = 30

wELCOME

This is the ANSWER

Someone help me pls

Answers

Answer:

-1 = 2(-1 -1) = -4

0 = 2(0-1) = -2

1 = 2(1-1) = 0

2 = 2(2-1) = 2

Step-by-step explanation:

x = the number on the left column

for the middle colum, you need to do 2(x -1)

for the right column, you do : (x - 1) x 2)

what are the solutions to the system

Answers

[tex]10+y=5x+x^2~~~~.....(i)\\\\5x+y =1~~~....(ii)\\\\\text{From (i):}\\\\10+y = 5x +x ^2\\\\\\\implies y = 5x +x ^2 -10\\\\\text{Substitute y in equation (ii):}\\\\5x+5x+x^2-10=1\\\\\implies x^2 + 10x -11 =0\\\\\implies x^2 +11x -x -11=0\\\\\implies x(x+11) -(x+11) =0\\\\\implies (x-1)(x+11) =0\\\\\implies x =1 ~~ \text{or}~~ x = -11\\\\\text{Substitute x =~1 in equation (ii):}\\ \\5+y =1 \implies y =-4[/tex]

[tex]\\\text{Substitute}~ x =-11~ \text{in equation (ii):}\\\\\ 5(-11) + y = 1 \implies y = 1 +55 = 56 \\\\\text{Hence} (x,y) = (-11,56) ~~\text{and}~~ (x,y)=(1,-4)[/tex]

Hasan plans to choose a book from a section of the store where everything is 25%
off He writes the expression d 0.25d to find the sale price of the book if the original
price is d dollars.
Bella correctly writes another
expression. 0.75d, that will also find the sale price of the
book if the original price is
I/ dollars,
Use the drop-down menus to explain what each part of Hasan's and Bella's
expressions mean.

Answers

Answers:

Hasan's Expression: d - 0.25d

d: Original price0.25: percent discount 0.25d: amount of discountd - 0.25d: sale price

--------------------

Bella's Expression: 0.75d

0.75: percent of original priced: original price0.75d: sale price

=============================================

Explanation:

The variable d stands for the original price. The store is offering a 25% sale, which means that Hasan will save 25% of the original price. This means 0.25d represents the amount he saves (aka the discount amount).

For instance, let's say d = 100 is the original price. 25% of this is 0.25d = 25 dollars. So if the book is originally $100, then he saves $25.

As you can probably guess, the 0.75 comes from 1 - 0.25 = 0.75; it indicates that if Hasan saves 25%, then he still has to pay the remaining 75%

Notice how d - 0.25d = 1d - 0.25d = (1-0.25)d = 0.75d which points to the sale price being 75% of the original price d.

1
4
3
Time (h)
2
.
3
Cars
Washed
6
9
12
a. Proportional?
b. Equation:
C. Number of hours:
3. Cars washed:

Answers

Step-by-step explanation:

Proportional = time/cars washed

= 1/3

no. of hours= 90hours

cars washed = 30 cars washed

Helps me please help help me

Answers

Answer:

(2, -6)

Step-by-step explanation:

1. The rule for reflecting a point across the y-axis is (x, y) -> (-x, y)

This means that when reflecting across the y-axis, the x-coordinate changes signs.[tex](-(-2),-6)[/tex] [tex](2,-6)[/tex]

Therefore, the ordered pair that describes the location of C' is (2, -6).

Answer:

(2, -6)

Step-by-step explanation:

Whose statement is correct based on the expected value?

Answers

Step-by-step explanation:

only Hadley. and even that is not fully correct.

the expected value here gives us the overall most likely outcome of the scenario when throwing 2 balls.

and not ball by ball. and not directly a probability.

like rolling a die : the expected value of rolling a 6 in 6 attempts is 1 due to the individual roll probability of 1/6.

but the probabilty is NOT 1 or 100% (a guaranteed 6 in 6 attempts).

and that brings me to my concern with Hadley's statement. it says with conviction "the player will make ...". no, only "... will likely make ..." or "... on average".

Answer:

Step-by-step explanation:

neither statement is correct. right answer on khan

Other Questions
what unwinds DNA molecule Which sentence below has a misplaced modifier According to VSEPR theory, a molecule with the general formula AX2E2 (A is the central atom, X is the bonded atom and E is a lone pair of electrons) has how many regions of electron density around the central atom [ Select ] The flywheel on a steam engine rotates 60 revolutions every 15 seconds. What is the flywheel's rate of revolutions per second? Can you help me with this spanish DO AGREE THAT MOST CRIMINAL IN PRISON TODAY ARE WITH LOW LEVEL OF EMOTIONAL INTELLIGENCE?WHY? Which action was considered legal under the slave codes? Enslaved people could purchase property. An enslaved person could testify against a white person in court. An enslaved person could be granted freedom by the state government.Enslaved people could teach one another to read. The total length of 5 belts and 3 shoe laces is 4.2 m. The total length of 3 shoe laces is equal to the length of 2 beltsa) Find the length of a beltb) Find the total length of 7 belts and 2 shoe laces What are MAD (mitotic arrest deficient) proteins and when are they important in the cell cycle? Brea and colin do business as dig excavators. In acting on the firms behalf, brea makes an honest error in underestimating the cost of a certain project. In this situation, to her firm, brea is liable for a breach of the duty of. A rope is 225 centimeters long.You need the rope to be 1.5 meters long.How many centimeters should you cut off? Wind forms desert pavement through ___.A. sublimationB. random acts of kindnessC. depositionD. erosion Need all of these answers please describe miss maudie atkinson. how typical is she of maycombs women? what do the children think of her? Please help !!!Will give the brainliest..Please answer correctly.. What are three advantages of digital x-rays over film. universalizing religion definition ap human geography Quadrilateral BCHS is a parallelogram with diagonal BHintersecting diagonal CS at P.angle BCH = 3x-3.86angle CHS = 2x+6.58Find the measure of angle HSB System of equations pls help me After watching some fish 55 feet below the surface of the water, a scuba diver went up 25 feet to explore a coral reef. Use a number line to help you create an equation that shows the location of the coral reef in relation to the water's surface. It might be A I really dont know lol