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
natali 33 [55]
3 years ago
5

Use a surface integral to find the surface area of the portion of the sphere xUse a surface integral to find the surface area of t

he portion of the sphere x^2 +y^2 +z^2 = 1 inside the cone z^2 = x^2 + y^2 above the xy-plane.
2 +y2 +z2 = 1 inside the cone z2 = x2 + y2 above the xy-plane.
Mathematics
1 answer:
brilliants [131]3 years ago
4 0

Parameterize this surface (call it S) by

\vec r(u,v)=\cos u\sin v\,\vec\imath+\sin u\sin v\,\vec\jmath+\cos v\,\vec k

with 0\le u\le2\pi and 0\le v\le\dfrac\pi4. The limits on u should be obvious. We find the upper limit for v by solving for v where the sphere and cone intersect:

\begin{cases}x^2+y^2+z^2=1\\z=\sqrt{x^2+y^2}\end{cases}\implies x^2+y^2=\dfrac12

\implies(\cos u\sin v)^2+(\sin u\sin v)^2=\dfrac12

\implies\sin^2v=\dfrac12

\implies\sin v=\dfrac1{\sqrt2}\implies v=\dfrac\pi4

Take the normal vector to S to be

\vec r_u\times\vec r_v=-\cos u\sin^2v\,\vec\imath-\sin u\sin^2v\,\vec\jmath-\cos v\sin v\,\vec k

(orientation does not matter here)

Then the area of S is

\displaystyle\iint_S\mathrm dA=\iint_S\|\vec r_u\times\vec r_v\|\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^{\pi/4}\int_0^{2\pi}\sin v\,\mathrm du\,\mathrm dv

=\displaystyle2\pi\int_0^{\pi/4}\sin v\,\mathrm dv=\boxed{(2-\sqrt2)\pi}

You might be interested in
Which expression is the factored form of 2/3x+4 ?
frosja888 [35]

Answer:

Option C.

Step-by-step explanation:

The given expression is

\dfrac{2}{3}x+4

We need to find the factored form of the given expression.

The given expression can be rewritten as

\dfrac{2}{3}x+(\dfrac{2}{3}\times \dfrac{3}{2})\times 4

\dfrac{2}{3}x+\dfrac{2}{3}(6)

It is clear that \frac{2}{3} is common.

Taking out common factors, we get

\dfrac{2}{3}(x+6)

It is the factored form of the given expression.

Hence, the correct option is C.

7 0
3 years ago
Read 2 more answers
If r is the radius of a circle and d is its diameter which of the following is an equivalent formula for the circumference c = 2
Ratling [72]

Answer:

C

Step-by-step explanation:

C=2pier or pied

8 0
3 years ago
Read 2 more answers
Please help. 10 points and brainliest.
Snezhnost [94]

Answer:

\frac{a^2}{b^2}

Step-by-step explanation:

Apply the exponent rule a^b*a^c=a^b+c to a

a^4*a^-^2=a^4^-^2=a^2

b^3a^2b^-^5

Apply the exponent rule to b

b^3*b^-^5=b^3^-^5=b^-^2

Apply the exponent rule a^-^b=\frac{1}{a^b}

b^-^2=\frac{1}{b^2}=\frac{1}{b^2}a^2=\frac{a^2}{b^2}

6 0
3 years ago
Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 11 in. by 7 in. by
11111nata11111 [884]

Answer:

The dimension of the open rectangular box is 8.216\times 4.216\times 1.392.

The volume of the box is 8.217 cubic inches.

Step-by-step explanation:

Given : The open rectangular box of maximum volume that can be made from a sheet of cardboard 11 in. by 7 in. by cutting congruent squares from the corners and folding up the sides.

To find : The dimensions and the volume of the open rectangular box ?

Solution :

Let the height be 'x'.

The length of the box is '11-2x'.

The breadth of the box is '7-2x'.

The volume of the box is V=l\times b\times h

V=(11-2x)\times (7-2x)\times x

V=4x^3-36x^2+77x

Derivate w.r.t x,

V'(x)=4(3x^2)-2(36x)+77

V'(x)=12x^2-72x+77

The critical point when V'(x)=0

12x^2-72x+77=0

Solve by quadratic formula,

x=\frac{18+\sqrt{93}}{6},\frac{18-\sqrt{93}}{6}

x=4.607,1.392

Derivate again w.r.t x,

V''(x)=24x-72

Now, V''(4.607)=24(4.607)-72=38.568>0 (+ve)

V''(1.392)=24(1.392)-72=-38.592 (-ve)

So, there is maximum at x=1.392.

The length of the box is l=11-2x

l=11-2(1.392)=8.216

The breadth of the box is b=7-2x

b=7-2(1.392)=4.216

The height of the box is h=1.392.

The dimension of the open rectangular box is 8.216\times 4.216\times 1.392.

The volume of the box is V=l\times b\times h

V=8.216\times 4.216\times 1.392

V=48.217\ in.^3

The volume of the box is 8.217 cubic inches.

5 0
3 years ago
Find the principal square root of 14.
Anika [276]
Sqrt14 = 3.74 (2dp)

The principal root is always the positive.
5 0
2 years ago
Other questions:
  • On average, Sasha can drive 28 miles on every gallon of gasoline.
    5·1 answer
  • EASYY MATH QUESTIon50 POINTS
    15·2 answers
  • The volume of the cylinder below is 108π cm^3.
    10·1 answer
  • (being timed)
    9·2 answers
  • Who is similar to Giacomo Puccini now in modern-days? And what are their similarities?​
    14·1 answer
  • Please help me with this question :)​
    10·2 answers
  • Solve the equation.<br> m=
    11·2 answers
  • There are 221 students on a field trip.
    11·2 answers
  • (sin^2x) / (1-cosx) = (secx+1) / (secx)
    6·1 answer
  • 4l+8m=40 please. Trying to solve for l and m
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!