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irga5000 [103]
2 years ago
6

Use spherical coordinates to find the volume of the region that lies outside the cone z = p x 2 + y 2 but inside the sphere x 2

+ y 2 + z 2 = 2. Write the answer as an exact answer, which should involve π and √ 2. Do not round or use a calculator.

Mathematics
1 answer:
Harman [31]2 years ago
4 0

I assume the cone has equation z=\sqrt{x^2+y^2} (i.e. the upper half of the infinite cone given by z^2=x^2+y^2). Take

\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi

The volume of the described region (call it R) is

\displaystyle\iiint_R\mathrm dx\,\mathrm dy\,\mathrm dz=\int_0^{2\pi}\int_0^{\sqrt2}\int_{\pi/4}^\pi\rho^2\sin\varphi\,\mathrm d\varphi\,\mathrm d\rho\,\mathrm d\theta

The limits on \theta and \rho should be obvious. The lower limit on \varphi is obtained by first determining the intersection of the cone and sphere lies in the cylinder x^2+y^2=1. The distance between the central axis of the cone and this intersection is 1. The sphere has radius \sqrt2. Then \varphi satisfies

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

(I've added a picture to better demonstrate this)

Computing the integral is trivial. We have

\displaystyle2\pi\left(\int_0^{\sqrt2}\rho^2\,\mathrm d\rho\right)\left(\int_{\pi/4}^\pi\sin\varphi\,\mathrm d\varphi\right)=\boxed{\frac43(1+\sqrt2)\pi}

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sergij07 [2.7K]

Answer:

x=630

Step-by-step explanation:

The Exponential Product rule tells us that...

x^m*x^n=x^{m+n}

Therefore, x=1+2+3...+35.

Therefore, x=630

5 0
2 years ago
Find the radius of a sphere with a volume of 26, 2440 cubic inches
ehidna [41]

Answer:

8.7304 x 1010

Step-by-step explanation:

I think its that >_<

The last 10 is supposed to be on top to the right of the 10

4 0
3 years ago
ANSWER ASAP PLEASE!!!
ioda

Answer: 5/3

Step-by-step explanation:

(3/5)x = 12

(5/3)(3/5)x = (5/3)12

x = 20

Therefore, choose Option D: 5/3

Hope I helped! <3

4 0
3 years ago
The length of a rectangle is twice its width. if the perimeter of the rectangle is 48 yd , find its area.
Katarina [22]
X = width
2x = length

2x + 4x = 48
6x = 48
x = 8
2x = 16

8*16 = 128

Your answer should be that the area is 128 square yards. Hope this helps!
7 0
3 years ago
What is the function that defines the following sequence? 10, 12, 14, 16, 18…
Furkat [3]

The sequence shown is defined by a function that generates <em>even</em> numbers equal or greater than 10, defined by the function s = 10 + 2 · (n - 1).

<h3>How to define the function behind a sequence</h3>

Sequences are sets of elements characterized by at least a rule. In this case, the sequence shown is characterized by a function that generates even numbers equal or greater than 10. The function behind the sequence is shown below:

s = 10 + 2 · (n - 1)     (1)

Where n is the <em>element</em> index.

The sequence shown is defined by a function that generates <em>even</em> numbers equal or greater than 10, defined by the function s = 10 + 2 · (n - 1).

To learn more on sequences: brainly.com/question/21961097

#SPJ1

4 0
1 year ago
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