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
OlgaM077 [116]
2 years ago
14

Compute the surface area of the portion of the sphere with center the origin and radius 4 that lies inside the cylinder x^2+y^2=

12
Mathematics
1 answer:
Tom [10]2 years ago
4 0

Answer:

16π

Step-by-step explanation:

Given that:

The sphere of the radius = x^2 + y^2 +z^2 = 4^2

z^2 = 16 -x^2 -y^2

z = \sqrt{16-x^2-y^2}

The partial derivatives of Z_x = \dfrac{-2x}{2 \sqrt{16-x^2 -y^2}}

Z_x = \dfrac{-x}{\sqrt{16-x^2 -y^2}}

Similarly;

Z_y = \dfrac{-y}{\sqrt{16-x^2 -y^2}}

∴

dS = \sqrt{1 + Z_x^2 +Z_y^2} \ \ . dA

=\sqrt{1 + \dfrac{x^2}{16-x^2-y^2} + \dfrac{y^2}{16-x^2-y^2} }\ \ .dA

=\sqrt{ \dfrac{16}{16-x^2-y^2}  }\ \ .dA

=\dfrac{4}{\sqrt{ 16-x^2-y^2}  }\ \ .dA

Now; the region R = x² + y² = 12

Let;

x = rcosθ = x; x varies from 0 to 2π

y = rsinθ = y; y varies from 0 to \sqrt{12}

dA = rdrdθ

∴

The surface area S = \int \limits _R \int \ dS

=  \int \limits _R\int  \ \dfrac{4}{\sqrt{ 16-x^2 -y^2} } \ dA

= \int \limits^{2 \pi}_{0} } \int^{\sqrt{12}}_{0} \dfrac{4}{\sqrt{16-r^2}} \  \ rdrd \theta

= 2 \pi \int^{\sqrt{12}}_{0} \ \dfrac{4r}{\sqrt{16-r^2}}\ dr

= 2 \pi \times 4 \Bigg [ \dfrac{\sqrt{16-r^2}}{\dfrac{1}{2}(-2)} \Bigg]^{\sqrt{12}}_{0}

= 8\pi ( - \sqrt{4} + \sqrt{16})

= 8π ( -2 + 4)

= 8π(2)

= 16π

You might be interested in
The earth is 4,540,000,000 years old.
Ann [662]

Answer:10^8

Step-by-step explanation:

8 0
3 years ago
Help please. what does x equal.
Tresset [83]

The answer would be 53

4 0
2 years ago
Kelly leaves her home and cycles 4y miles south then cycles (3y + 9) miles east
atroni [7]
4y(3y + 9)
12y + 36
36 ÷ 12=x=3
7 0
3 years ago
The radius of the larger circle is 10 cm. The radius of the smaller circle is 3 cm. What is the approximate area of the shaded r
romanna [79]
I just took the test the correct answer is 285.74 square centimeters 
6 0
2 years ago
Using fraction busting solve for X <br><br>1/x² + 1/x = 6/x² ​
777dan777 [17]
The solution for this question would be. x += 5 hope it helps, and im truly am sorry if its incorrect!
8 0
2 years ago
Other questions:
  • Prove a/b is less than b/a when 0 &lt; a &lt;b
    15·1 answer
  • A microscope shows you an image of an object that is 20 times the objects actual size. Therefore, the scale factor of the enlarg
    12·2 answers
  • Can someone please help me ive been stuck in these questions for ages
    11·1 answer
  • What number is 120% of 20?
    7·2 answers
  • An archeology dig site is shaped like a square. The archeologists make a map of the
    13·2 answers
  • The perimeter of the triangle below is 10x - 3<br><br> Show how to solve
    13·1 answer
  • The width of a rectangle is 4y-2.5 feet and the length is 2.5y+10 feet. Find the perimeter of the rectangle.
    11·2 answers
  • Evaluate:<br> -22+(-45) = [?]
    8·2 answers
  • I need help w this pls
    5·1 answer
  • Help please and thank you.
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!