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
SpyIntel [72]
3 years ago
6

Use cylindrical coordinates to evaluate the triple integral ∭ where E is the solid bounded by the circular paraboloid z = 9 - 16

(x^2 + y^2) and the xy -plane.
Mathematics
1 answer:
4vir4ik [10]3 years ago
4 0

Answer:

\mathbf{\iiint_E  E \sqrt{x^2+y^2} \ dV =\dfrac{81 \  \pi}{80}}

Step-by-step explanation:

The Cylindrical coordinates are:

x = rcosθ, y = rsinθ and z = z

From the question, on the xy-plane;

9 -16 (x^2 + y^2) = 0 \\ \\  16 (x^2 + y^2)  = 9 \\ \\  x^2+y^2 = \dfrac{9}{16}

x^2+y^2 = (\dfrac{3}{4})^2

where:

0 ≤ r ≤ \dfrac{3}{4} and 0 ≤ θ ≤ 2π

∴

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0} \int ^{9-16r^2}_{0} \ r \times rdzdrd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0} r^2 z|^{z= 9-16r^2}_{z=0}  \ \ \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0} r^2 ( 9-16r^2})  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0}  ( 9r^2-16r^4})  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0}   ( \dfrac{9r^3}{3}-\dfrac{16r^5}{5}})|^{\dfrac{3}{4}}_{0}  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0}   ( 3r^3}-\dfrac{16r^5}{5}})|^{\dfrac{3}{4}}_{0}  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0}   ( 3(\dfrac{3}{4})^3}-\dfrac{16(\dfrac{3}{4})^5}{5}}) d \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV =( 3(\dfrac{3}{4})^3}-\dfrac{16(\dfrac{3}{4})^5}{5}}) \theta |^{2 \pi}_{0}

\iiint_E  E \sqrt{x^2+y^2} \ dV =( 3(\dfrac{3}{4})^3}-\dfrac{16(\dfrac{3}{4})^5}{5}})2 \pi

\iiint_E  E \sqrt{x^2+y^2} \ dV =(\dfrac{81}{64}}-\dfrac{243}{320}})2 \pi

\iiint_E  E \sqrt{x^2+y^2} \ dV =(\dfrac{81}{160}})2 \pi

\mathbf{\iiint_E  E \sqrt{x^2+y^2} \ dV =\dfrac{81 \  \pi}{80}}

You might be interested in
Which solid, no matter at which angle you cut it, will always have a circle for a cross section?
mestny [16]
Sphere because it is a 3D circle. will always have a circle if you cut it in half.


D) Sphere
6 0
3 years ago
Read 2 more answers
What is one 1/4 added to one 1/8
Ray Of Light [21]
1/4+1/8= 0.375 in decimals but in a fraction form is 3/8 hope it helps!
4 0
3 years ago
Read 2 more answers
Perform the following<br> f (x) = 4x2 + 5x + 5<br> g(x) = 8x + 6<br> Find (f + g)(x).
guapka [62]

Answer:

maya 6528

Step-by-step explanation:

$0rry so no alam

4 0
3 years ago
For question 9 and 10 use the graph below
miskamm [114]

Answer:the x intercept is (0,-7) that is number 9 and number 10 is (0,2)

Step-by-step explanation:

6 0
3 years ago
I don't know how to solve it . I don't know how to step it up
astraxan [27]
(-1,2) solve for the first variable in one of the equations, then substitute the result into the other equation
7 0
3 years ago
Other questions:
  • Brigid is picking strawberries at the Pick-Your-Own Farm. Her goal is to pick 5 bushels of strawberries. She has already picked
    6·2 answers
  • How many sequences of length 5 can be made when each component of the sequence can take 3 different values?
    6·1 answer
  • Help pls! Brainliest!
    6·1 answer
  • Write an equation of the line passing through the points ​(4​,11​) and ​(−1​,−14​).
    10·1 answer
  • Stan borrows $5,500 at a rate of 12% interest per year. What is the amount due at the end of 5 years if the interest is compound
    7·1 answer
  • I will give braliest to the person who finishes this first.
    13·2 answers
  • If sin f° = eight ninths and the measure of segment YW is 24 units, what is the measure of segment YX?
    8·1 answer
  • Functions A and B are shown in the equation and graph. select the true statement. ​
    15·1 answer
  • *please help*<br><br> I just need someone to check this for me.. am i doing this right?
    15·1 answer
  • If I have three crackers and I ate one how many crackers do I have now?
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!