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
AnnZ [28]
2 years ago
11

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 oct

ant. 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.)
Mathematics
1 answer:
irga5000 [103]2 years ago
3 0

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

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\}

and the integral of y over U is

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

\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

\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

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

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

\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)

\displaystyle \iiint_U y \, dV =  \pi - \sin(\pi) = \boxed{\pi}

You might be interested in
A game room has a floor that is 200’ x 20’ a scale drawing of the floor on grid paper using a scale of two units in 5 feet what
kondor19780726 [428]
Multiply it then see what 5 scales are
8 0
3 years ago
A shop has 15% sale. Originally a computer cost £275. How much does the computer cost in the sale?
lutik1710 [3]
15%×£275=£41.25
£275-£41.25=£233.75
5 0
3 years ago
Read 2 more answers
Round 7,954 to the place named.
Luden [163]

Answer:

8.0

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Please need help with this question anyone ?
Zina [86]

Answer:

106 = 1,000,000 x 3.63

Step-by-step explanation:

a = 3.63 (Please notice any zeroes on the end have been removed)

b = 6

3.63 x 106

Check your work:

106 = 1,000,000 x 3.63 = 3,630,000

4 0
3 years ago
First one to answer randomly gets brainliest!
s344n2d4d5 [400]

Answer:

,

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Other questions:
  • Solve the combined inequality and describe the graph of the solution.
    7·1 answer
  • Need help really fast please!!
    8·1 answer
  • Please help this is my last one
    6·1 answer
  • if you are dealt 3 cards from a shuffled deck of 52 cards find the probability that all 3 cards are picture cards
    13·1 answer
  • What is 3×584 in expanded form
    9·2 answers
  • What is -2 1/2 x -3 1/3
    14·1 answer
  • A rectangular field has an area of 2100 square feet. the length of the field is 50 feet.
    11·1 answer
  • can somebody please just explain to me how to do this, it says solve for x in each diagram. I already answered the first questio
    13·1 answer
  • The domain of y = cosx is the set of real numbers. True False
    9·1 answer
  • Solve for x.<br> -2-2x/3=-8<br> = [?]
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!