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
Someone help me plsh
Klio2033 [76]

Answer:

See attached image

Step-by-step explanation:

With this type of reflection, one has to convert the points of the form (x, y) into (y, x).

So the vertices of the triangle will transform the following way:

(4, 4)   into image (4, 4)

(4, 1)   into image  (1, 4)

(2, 1)  into image  (1, 2)

Please see attached picture to find the new reflected triangle depicted in red.

8 0
3 years ago
On awall of dimension10.5m long and 8.5 m wide, a square shaped wall posture is stuck at the centre whose one side measures2.5m.
tangare [24]

Answer:

₹996

Step-by-step explanation:

Area painted = area of wall - area of wall posture

Area of a rectangle = length x breadth

= 10.5m x 8.5m = 89.25m^2

Area of a square

length x length

2.5m x 2.5m = 6.25m^2

Area painted =  89.25m^2 -  6.25m^2 = 83m^2

Total cost of area painted = 83m^2 x 12 = 996

4 0
3 years ago
10.9 + (-15.6) + 2.1 write as a decimal​
Delvig [45]

Answer:

-2.6

Step-by-step explanation:

10.9 + (-15.6) becomes 10.9 - 15.6 which equals -4.7. The last step is to add the 2.1 to the -4.7: -4.7 + 2.1 = -2.6

Therefore the answer is -2.6

5 0
3 years ago
How do you write an equation for "a number decreased by negative 7 is the same thing as 48
just olya [345]
N-(-7)=48 I'm not sure though
3 0
3 years ago
Read 2 more answers
Find the product.<br><br> 2a2(5b2+3ab+6a+1)<br><br> PLEASE HELP!!!! ASAP!!!
Anon25 [30]

Answer:

10a^{2} b^{2} +6a^{3} b+12a^{3} +2a^{2}

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Other questions:
  • What is the equation of the function that is graphed as line b?
    13·2 answers
  • The equation y = 15x + 10 gives your score on a math quiz, where x is the number of questions you answered correctly. What is yo
    7·2 answers
  • Explain why 1/12+1/12+1/12 is the same as 1/4
    14·1 answer
  • What is the value of the expression [(29+18)+(17-8)]÷8
    10·1 answer
  • What is the value of x?<br> In a circle
    14·1 answer
  • 6. A leather bag is bought for Rs 18000 and sold with 15% loss. Find the selling price of the bag.​
    6·2 answers
  • 10+10+10+10+10+10+10+10
    8·2 answers
  • HELPPPPP PLSSSS!!!! You start at (-2, 3). You move up 1 unit and right 3 units. Where do you end? sp 4 3 2 1 X -5 -4 -3 -2 -1 0
    10·2 answers
  • Jamie's family has completed 80% of a trip. They have traveled 40 miles. How far is the trip?​
    14·1 answer
  • Caitlyn pays $2.20 in postage to mail a CD to a friend. She uses 43-cent stamps and 6-cent stamps. How many of each stamp did Ca
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!