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3 years ago

Someone please help me on this work it would be very helpful for me

Mathematics

1 answer:

Blababa [14]3 years ago

4 0

Sorry i don’t know:(

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Rewrite the following integral in spherical coordinates.

lora16 [44]

In cylindrical coordinates, we have $r^2=x^2+y^2$ , so that

$z = \pm \sqrt{2-r^2} = \pm \sqrt{2-x^2-y^2}$

correspond to the upper and lower halves of a sphere with radius $\sqrt2$ . In spherical coordinates, this sphere is $\rho=\sqrt2$ .

$1 \le r \le \sqrt2$ means our region is between two cylinders with radius 1 and $\sqrt2$ . In spherical coordinates, the inner cylinder has equation

$x^2+y^2 = 1 \implies \rho^2\cos^2(\theta) \sin^2(\phi) + \rho^2\sin^2(\theta) \sin^2(\phi) = \rho^2 \sin^2(\phi) = 1 \\\\ \implies \rho^2 = \csc^2(\phi) \\\\ \implies \rho = \csc(\phi)$

This cylinder meets the sphere when

$x^2 + y^2 + z^2 = 1 + z^2 = 2 \implies z^2 = 1 \\\\ \implies \rho^2 \cos^2(\phi) = 1 \\\\ \implies \rho^2 = \sec^2(\phi) \\\\ \implies \rho = \sec(\phi)$

which occurs at

$\csc(\phi) = \sec(\phi) \implies \tan(\phi) = 1 \implies \phi = \dfrac\pi4+n\pi$

where $n\in\Bbb Z$ . Then $\frac\pi4\le\phi\le\frac{3\pi}4$ .

The volume element transforms to

$dx\,dy\,dz = r\,dr\,d\theta\,dz = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi$

Putting everything together, we have

$\displaystyle \int_0^{2\pi} \int_1^{\sqrt2} \int_{-\sqrt{2-r^2}}^{\sqrt{2-r^2}} r \, dz \, dr \, d\theta = \boxed{\int_0^{2\pi} \int_{\pi/4}^{3\pi/4} \int_{\csc(\phi)}^{\sqrt2} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta} = \frac{4\pi}3$

4 0

2 years ago

What is the answer for number 7?

aivan3 [116]

Answer:

Step-by-step explanation:

x=30

m=abe=180

8 0

3 years ago

PLEASE HELP 4. The Buff City Soaps in Arlington is counting how many soaps and bath bombs they sell in a given day. In total, th

Vedmedyk [2.9K]

Answer:

33 = x + (3x + 1)

x = bath bombs

So 3x is three times as many bath bombs, then the plus one makes it so that the soaps sold is more than three times the amount of bath bombs sold

8 0

3 years ago

The average income, I, in dollars, of a lawyer with an age of x years is modeled with the following function:

Alinara [238K]

The answer is 37 years old.

I = -425x² + 45500x -650000
I = 450000

-425x² + 45500x -650000 = 450000
-425x² + 45500x - 650000 - 450000 = 0
-425x² + 45500x - 1100000 = 0

Divide all factors by 25:
- 17x² + 1820x - 44000 = 0

This is quadratic equation (ax² + bx + c = 0)
$x_{1,2} = \frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} = \frac{-1820+/- \sqrt{ 1820^{2}-4*(-17)*(-44000) } }{2*(-17)} = \\ \\ = \frac{-1820+/- \sqrt{3312400-2992000 } }{-34} = \frac{-1820+/- \sqrt{320400} }{-34} =\frac{-1820+/- 566}{-34} \\ \\ 
x_1= \frac{-1820+ 566}{-34}= -37 \\ \\ 
x_2 = \frac{-1820- 566}{-34}=70$

Since, the youngest age is needed, the correct answer is x = 37

3 0

3 years ago

Felicia places a rectangular rug that measures 10 feet by 11 feet in a triangular room with a base of 30 feet and height of 22 f

arsen [322]

Answer:

1. Area of rectangular rug is $110 ft^{2}$