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
Is (4, -2) the solution of the system of equations shown
Advocard [28]

No, because if you substitute (4, -2) into x and y in each equation, they are not the same as the value on the right side of the equation.

2(4) + 4(-2) = - 2

8 + -8 = -2

0 = -2 <---- this is not correct.

-4(4) + -2 = -14

-16 + -2 = -14

-18 = -14 <----- this is not correct.

8 0
3 years ago
George needs blueberries for a certain recipe calls for 4/5 cup blueberries.However, George wants to make a recipe that is 3 1/2
lesya [120]

Answer:

Step-by-step explanation:

4/5 × 3 1/2

= 4/5 × 7/2

= 28/10

= 2 8/10

=2 4/5 cups of blueberries

6 0
3 years ago
Find the slope from the pair of points.<br> (0, -5) (-3,5)
bagirrra123 [75]

Answer:

Step-by-step explanation:

your welcome

5 0
3 years ago
Who can help me needed for tomorrow please
blagie [28]

a) 170.00 =8n -30

200= 8n

n = 200/8 = 25 items


b) since the expenses increased the profits would decrease

profit = 8(25)-40 = 200-40 = 160

170-160 = 10.00

 profit decreased by $10.00

6 0
3 years ago
Read 2 more answers
. Jessica went to the grocery store and purchased a box of cereal for $3.75, milk for $1.50, and bread for $2.25. She paid with
puteri [66]
$3.75 - cereal
$1.50 - milk
$2.25 - bread
—————————
$7.50

$20.00
- 7.50
—————
$ 12.50


She received $12.50 back
4 0
3 years ago
Other questions:
  • Morgan plots the point (4,-7) on a coordinate plane. If she reflects the point aacross the y axis, what are the coordinates of t
    11·2 answers
  • X + 12/x + 2 = x/x + 8<br> A) X = -4<br> B) X = 8<br> C) X = 6<br> D) X = -6
    9·1 answer
  • Largest to smallest 1\1\4
    13·2 answers
  • Help me plz show work
    12·1 answer
  • 5/8 divided by five and three fourth
    12·1 answer
  • I need help I just started this
    7·2 answers
  • O 6) Missy walks 1/3 mile to school Select all the fractions that are equivalent to 1/3. 1/6 2/6 4/12 3/6 3/4 This is a required
    11·1 answer
  • Heather has an international cell phone plan.
    6·2 answers
  • Ohio City Burgers charges $6.75 for a
    15·1 answer
  • I neeeeeed help pls 6th grade easy
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!