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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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IrinaK [193]
<h3>Answer:  (a - 1)(2a² + a + 2)</h3>

=========================================================

Explanation:

Use the rational root theorem to determine this list of possible rational roots: 1, -1, 1/2, -1/2

Plug each possible root one at a time into the original expression given. If the simplified result is 0, then that possible root is an actual root.

If we tried say a = -1, then,

2a³-a²+a-2​ = 2(-1)³-(-1)²+(-1)-2​ = -6

The result is not zero, so a = -1 is not an actual root.

But if we tried say a = 1, then,

2a³-a²+a-2​ = 2(1)³-1²+1-2​ = 0

We get 0 so a = 1 is an actual root. I'll let you try the other values, but you should find that a = 1 is the only rational root.

Since a = 1 is a root, this makes (a-1) to be a factor.

From here, use either synthetic or polynomial long division to determine the other factor. Refer to the diagram below for each method.

Regardless of which method you pick, the quotient is 2a² + a + 2 which is the other factor needed. The remainder of 0 tells us we have (a-1) as a factor. For more information, check out the remainder theorem.

5 0
1 year ago
Find the fifth term in the geometric sequence for a1=2 and r=3
leonid [27]

D=4Step-by-step explanation:

6 0
3 years ago
Half-life of Po-210 is 140 days. If the initial mass of the sample is 5
Anna35 [415]

Answer:

0.625kg

Step-by-step explanation:

we are given half-life of PO-210 and the initial mass

we want to figure out the remaining mass <u>after</u><u> </u><u>4</u><u>2</u><u>0</u><u> </u><u>days</u><u> </u>

in order to solve so we can consider the half-life formula given by

\displaystyle f(t) = a {0.5}^{t/T}

where:

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since it halves every 140 days our T is 140 and t is 420. as the initial mass of the sample is 5 our a is 5

thus substitute:

\displaystyle f(420)=5\cdot{0.5}^{420/140}

reduce fraction:

\displaystyle f(420)=5\cdot{0.5}^{3}

By using calculator we acquire:

\displaystyle f(420)=0.625

hence, the remaining sample after 420 days is 0.625 kg

4 0
3 years ago
Find one of the solutions to the following absolute value equation:<br> 26 + 4) = 48
Sindrei [870]
Answer : MAYBE 18 (?)
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30 = 48
Then ( this is the part I’m not sure about)
You subtract 30 out that means 30-30=0 & 48-30 = 18 so that’s where I found my answer but idk I haven’t done absolute value since Algebra 1 !!
7 0
3 years ago
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aalyn [17]
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6.8 x 10^13  *  1 light year / 9.46 x 10^12 = 7.188 light years.


3 0
3 years ago
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