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antiseptic1488 [7]
2 years ago
12

Find the volume of the solid formed by rotating the region bounded by the given curves about the indicated axis. Y = 1x, y = 1,

y = 5, x = 0; about the y-axis.
Mathematics
1 answer:
Goryan [66]2 years ago
8 0

The disk method will only involve a single integral. I've attached a sketch of the bounded region (in red) and one such disk made by revolving it around the y-axis.

Such a disk has radius x = 1/y and height/thickness ∆y, so that the volume of one such disk is

π (radius) (height) = π (1/y)² ∆y = π/y² ∆y

and the volume of a stack of n such disks is

\displaystyle V_n = \sum_{i=1}^n \pi {y_i}^2 \Delta y

where y_i is a point sampled from the interval [1, 5].

As we refine the solid by adding increasingly more, increasingly thinner disks, so that ∆y converges to 0, the sum converges to a definite integral that gives the exact volume V,

\displaystyle V = \lim_{n\to\infty} V_n = \int_1^5 \frac{\pi}{y^2} \, dy

V = -\dfrac\pi y\bigg|_{y=1}^{y=5} = \boxed{\dfrac{4\pi}5}

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Oduvanchick [21]

Answer:

Once. at (3,0)

Step-by-step explanation:

It is handy to have a graphing calculator for this kind of question.

Otherwise work it out.

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then factor (x -3)(x-3)

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7 0
3 years ago
The verticle of a triangle are (2, 18) (-2, -4), and (6,12.
Sergeu [11.5K]

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(2, 18) , (-2, -4)

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(6,12) , (2, 18)


\bf (\stackrel{x_1}{2}~,~\stackrel{y_1}{18})\qquad (\stackrel{x_2}{-2}~,~\stackrel{y_2}{-4}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{-4-18}{-2-2}\implies \cfrac{-22}{-4}\implies \cfrac{11}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-18=\cfrac{11}{2}(x-2) \\\\\\ y-18=\cfrac{11}{2}x-11 \implies \blacktriangleright y=\cfrac{11}{2}x+7 \blacktriangleleft


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) The density of oil in a circular oil slick on the surface of the ocean at a distance of r meters from the center of the slick
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Answer:

Therefore the mass of the of the oil is 409.59 kg.

Step-by-step explanation:

Let us consider a circular disk. The inner radius of the disk be r and the outer diameter of the disk be (r+Δr).

The area of the disk

=The area of the outer circle - The area of the inner circle

= \pi (r+\triangle r)^2- \pi r^2

=\pi [r^2+2r\triangle r+(\triangle r)^2-r^2]

=\pi [2r\triangle r+(\triangle r)^2]

Since (Δr)² is very small, So it is ignorable.

∴A=2\pi r\triangle r

The density \delta (r)= \frac{40}{1+r^2}

We know,

Mass= Area× density

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Total mass M=\sum_{i=1}^n \frac{80r_i\pi }{1+r^2}\triangle r_i

Therefore

\sum_{i=1}^n \frac{80r_i\pi }{1+r^2}\triangle r_i=\int_0^5 \frac{80r\pi }{1+r^2}dr

                      =40\pi[ln(1+r^2)]_0^5

                      =40\pi [ln(1+5^2)-ln(1+0^2)]

                     =40\pi ln(26)

                     = 409.59 kg (approx)

Therefore the mass of the of the oil is 409.59 kg.

3 0
2 years ago
Seven - twelfths of a new dress has been sewn. Chelsea did 3/7 of the sewing. How much of the dress did Chelsea sew?
galina1969 [7]
1/4, because 7/12×3/7=1/4
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