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Andru [333]
3 years ago
5

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by y=x^8 and y = 256

in the first quadrant about the y-axis.
Mathematics
1 answer:
Mkey [24]3 years ago
8 0

Using the shell method, the volume integral would be

\displaystyle 2\pi \int_0^2 x(256-x^8)\,\mathrm dx

That is, each shell has a radius of <em>x</em> (the distance from a given <em>x</em> in the interval [0, 2] to the axis of revolution, <em>x</em> = 0) and a height equal to the difference between the boundary curves <em>y</em> = <em>x</em> ⁸ and <em>y</em> = 256. Each shell contributes an infinitesimal volume of 2<em>π</em> (radius) (height) (thickness), so the total volume of the overall solid would be obtained by integrating over [0, 2].

The volume itself would be

\displaystyle 2\pi \int_0^2 x(256-x^8)\,\mathrm dx = 2\pi \left(128x^2-\frac1{10}x^{10}\right)\bigg|_{x=0}^{x=2} = \boxed{\frac{4096\pi}5}

Using the disk method, the integral for volume would be

\displaystyle \pi \int_0^{256} \left(\sqrt[8]{y}\right)^2\,\mathrm dy = \pi \int_0^{256} \sqrt[4]{y}\,\mathrm dy

where each disk would have a radius of <em>x</em> = ⁸√<em>y</em> (which comes from solving <em>y</em> = <em>x</em> ⁸ for <em>x</em>) and an infinitesimal height, such that each disk contributes an infinitesimal volume of <em>π</em> (radius)² (height). You would end up with the same volume, 4096<em>π</em>/5.

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Please help thanks thanks thanks ty
Archy [21]

Answer:

104 degrees

Step-by-step explanation:

First let's start by finding Angle DGC. Since Angle FGD and DGC are a linear pair, Angle DGC = 180-90 = 90 degrees.

Next we need to find GDC

Angles in a triangle add up to 180 degrees therefore...

Angle GDC + DCG + DGC = 180

Plug in the values we found into the equation

Angle GDC + 37 + 90 = 180

Angle GDC + 127 = 180

Angle GDC = 53

Therefore Angle ADC = 53 + 51 = 104 degrees

Since ABCD is a parallelogram, then opposite angles are equal therefore...

Angle B = 104 degrees

3 0
2 years ago
Please go to my profile and answer my last question
Alex Ar [27]

Answer:

ok

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
What is the value is of x?<br><br> A)0<br><br> B)25<br><br> C)4<br><br> D)7
iogann1982 [59]

We know that these two angles are equal to each other (There is the "congruent" sign) so we can set them equal to each other and solve for x

3x - 17 = 25 - 3x

(3x + 3x) - 17 = 25 + (-3x + 3x)

6x + (- 17 + 17) = 25 + 17

6x/6 = 42/6

x = 7

Hope this helped!

7 0
2 years ago
I need help im gonna post 3 more different questions pls answer them.
Arisa [49]

Answer:

15

Step-by-step explanation:

x³ - (3 + x)²

4³ - (3 + 4)²

8 0
3 years ago
Does anyone know the answers? I'm awful at evaluation. <br> |7-9|=<br> |7|-9=
ella [17]

Answer:

2; -2

Step-by-step explanation:

|7-9| = |-2| = 2

|7|-9= 7 - 9 = -2

3 0
3 years ago
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