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pishuonlain [190]
3 years ago
13

Y=x, y=0, y=4, x=6 : find the volume of the solid generated revolving around line x=6

Mathematics
1 answer:
Vilka [71]3 years ago
5 0
Using shells, the volume is

\displaystyle2\pi\int_0^6(6-x)x\,\mathrm dx

where 6-x is the distance from any point in the region along the x-axis to the axis of revolution x=6, which makes up the radius of each shell; and x is the height of each shell, which is determined by the line y=x.

So the volume is

\displaystyle2\pi\int_0^6(6x-x^2)\,\mathrm dx=2\pi\left(3x^2-\frac{x^3}3\right)\bigg|_{x=0}^{x=6}=72\pi
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A particle begins at point (1, 2) and is moving along the line segment joining (1, 2) to (3, 4). Initially, what is the rate of
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Answer:

Initially, that is, at (1,2), the rate of increase of the function in the direction given = 0

Hence the function is neither increasing nor decreasing initially in the given direction.

Step-by-step explanation:

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Rate of Change of the function = ∇f = (fₓ, fᵧ)

And we're told to find the rate of change in a particular direction = ∇f.û

We first obtain the unit vector in that direction, û

Direction = (3,4) - (1,2) = (2,2)

Uni vector = vector/magnitude

Vector = 2î + 2j

magnitude = √(2² + 2²) = √8 = 2 √2

(2î + 2j)/(2√2) = (1/√2)î + (1/√2)j = û

Unit vector in the direction = (1/√2, 1/√2)

f = 2x² - y²

At (1,2)

fₓ = ∂f/∂x = 4x = 4

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∇f = (4,-4)

∇f.û = (4î - 4j).((1/√2)î + (1/√2)j) = (4/√2) - (4/√2) = 0

If it was positive, then the function is increasing, if it was negative, the function is decreasing. Zero means neither of them.

Hence the function is neither increasing nor decreasing initially in the given direction.

7 0
2 years ago
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Answer:

90

Step-by-step explanation:

Angle Formed Outside = 1/2(DIFFERENCE of Intercepted Arcs)

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