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
disa [49]
1 year ago
8

Find the volume of the solid.

Mathematics
1 answer:
dmitriy555 [2]1 year ago
3 0

In Cartesian coordinates, the region (call it R) is the set

R = \left\{(x,y,z) ~:~ x\ge0 \text{ and } y\ge0 \text{ and } 2 \le z \le 4-x^2-y^2\right\}

In the plane z=2, we have

2 = 4 - x^2 - y^2 \implies x^2 + y^2 = 2 = \left(\sqrt2\right)^2

which is a circle with radius \sqrt2. Then we can better describe the solid by

R = \left\{(x,y,z) ~:~ 0 \le x \le \sqrt2 \text{ and } 0 \le y \le \sqrt{2 - x^2} \text{ and } 2 \le z \le 4 - x^2 - y^2 \right\}

so that the volume is

\displaystyle \iiint_R dV = \int_0^{\sqrt2} \int_0^{\sqrt{2-x^2}} \int_2^{4-x^2-y^2} dz \, dy \, dx

While doable, it's easier to compute the volume in cylindrical coordinates.

\begin{cases} x = r \cos(\theta) \\ y = r\sin(\theta) \\ z = \zeta \end{cases} \implies \begin{cases}x^2 + y^2 = r^2 \\ dV = r\,dr\,d\theta\,d\zeta\end{cases}

Then we can describe R in cylindrical coordinates by

R = \left\{(r,\theta,\zeta) ~:~ 0 \le r \le \sqrt2 \text{ and } 0 \le \theta \le\dfrac\pi2 \text{ and } 2 \le \zeta \le 4 - r^2\right\}

so that the volume is

\displaystyle \iiint_R dV = \int_0^{\pi/2} \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta \, dr \, d\theta \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta\,dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} r((4 - r^2) - 2) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} (2r-r^3) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \left(\left(\sqrt2\right)^2 - \frac{\left(\sqrt2\right)^4}4\right) = \boxed{\frac\pi2}

You might be interested in
A total of 160 children play in the city soccer league and 64 of the players are girls what percent of the players are girls
monitta

Answer:

40 percent of the children that play soccer are girls in the league

Step-by-step explanation:

just divide 64/160 and get 40

6 0
3 years ago
Read 2 more answers
Tyler’s brother earns a commission. He makes 2.5% of the amount he sells. Last week, he sold $900 worth of shoes. How much was h
lukranit [14]

Answer:

$22.5

Step-by-step explanation:

2.5% of 900

0.025(900)= 22.5

7 0
3 years ago
In a triangle, the largest angle is always opposite the (Blank) and the smallest angle is always opposite the (Blank) .
Shtirlitz [24]

Answer:

the largest angle is opposite the largest side and the shortest angle is opposite the shortest side

Step-by-step explanation:

7 0
3 years ago
The library Survey 240 people about their favorite type of movie is 50% of the people chose documentaries how many people chose
nevsk [136]
G. 36 people. There were only 36 people choose documentaries.
5 0
3 years ago
NEED HELP!!!!!!!!!!!!!!!!!!!!!
Likurg_2 [28]
I think the answer is C not to sure
3 0
3 years ago
Read 2 more answers
Other questions:
  • Can anyone explain to me what this is asking/how to do this? I think it's for the summation formula <img src="https://tex.z-dn.n
    13·1 answer
  • David writes down the sequence 2,6,10,14 he says the sequence is n+4 is he correct
    6·2 answers
  • The graph shows the number of patients seen by a doctor on Tuesday.
    14·1 answer
  • A master electrician earns $62 per hour. His apprentice earns $40 per hour. The master electrician works 3 hours more than the a
    9·2 answers
  • Which system is equivalent to StartLayout Enlarged left-brace 1st row y = 9 x squared 2nd row x + y = 5 EndLayout StartLayout En
    8·2 answers
  • How many extraneous solutions does the equation below have?
    15·2 answers
  • 5^2(5^3⋅5^2) <br> What is the answer
    12·1 answer
  • Solve for x.
    11·1 answer
  • Solve for the missing angle with the question mark
    7·1 answer
  • HELP PLEASE I NEED HELP ASAP!!
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!