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seraphim [82]
3 years ago
7

Find the exact area of the surface obtained by rotating the curve about the x-axis.

Mathematics
1 answer:
padilas [110]3 years ago
5 0
\bf \begin{cases}
y=sin\left( \frac{\pi x}{6} \right)\\
0\le x\le 6\\
\textit{about the x-axis, or }y=0
\end{cases}\\\\
-----------------------------\\\\
\textit{using the disc method}
\\\\
V=\int\limits_{0}^{6}\pi \left[ sin\left( \frac{\pi x}{6} \right) \right]^2\cdot dx\implies \pi\int\limits_{0}^{6} \left[ sin\left( \frac{\pi x}{6} \right) \right]^2\cdot dx
\\\\\\
 \pi\int\limits_{0}^{6} sin^2\left( \frac{\pi x}{6} \right) \cdot dx\\\\
-----------------------------\\\\

\bf \textit{now, let us check the double angle identities}
\\\\
cos(2\theta)=
\begin{cases}
cos^2(\theta)-sin^2(\theta)\\
\boxed{1-2sin^2(\theta)}\\
2cos^2(\theta)-1
\end{cases}
\\\\\\
thus\implies cos(2\theta)=1-2sin^2(\theta)\implies 2sin^2(\theta)=1-cos(2\theta)
\\\\\\
sin^2(\theta)=\cfrac{1-cos(2\theta)}{2}\qquad thus\\\\
-----------------------------

\bf \pi\int\limits_{0}^{6} \left[ sin\left( \frac{\pi x}{6} \right) \right]^2\cdot dx\implies 
\pi\int\limits_{0}^{6} \cfrac{1-cos\left(2\cdot  \frac{\pi x}{6} \right)}{2}\cdot dx
\\\\\\
\pi\int\limits_{0}^{6}\cfrac{1}{2}dx-\pi \cdot \cfrac{1}{2}\pi\int\limits_{0}^{6}cos\left(\frac{\pi x}{3} \right)dx
\\\\\\
\left[\cfrac{\pi x}{2}-\cfrac{\pi }{2}\cdot \cfrac{sin\left(\frac{\pi x}{3} \right)}{\frac{\pi x}{3} }  \right]\implies \left[ \cfrac{\pi }{2}x-\cfrac{3sin\left(\frac{\pi x}{3} \right)}{2x} \right]_0^6

and surely, you'd know how to get the values for the bounds there

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gulaghasi [49]
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8 0
3 years ago
B Write the coordinate of T after a dilation with a scale factor of 1/4 centered at the origin. 1 10 T х -10 0 S U 10 Write your
Vanyuwa [196]

Answer:

(4, 4)

Step-by-step explanation:

Coordinates of a point (x, y) after dilation with a scale factor k,

Rule to be followed,

(x, y) → k(x, y)

        → (kx, ky)

Coordinates of point T are (8, 8).

Dilation of the point T after dilation with a scale factor of \frac{1}{4} will be,

(8, 8) → \frac{1}{2}(8, 8)

        → (4, 4)

Therefore, coordinates of the image point B' will be (4, 4).

3 0
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natima [27]

Answer:

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Step-by-step explanation:

Let the width be w

so Length will be 2w

using the rectangle formula:

\sf Perimeter \ of \ rectangle= 2L + 2W

\sf 60= 2(2w) + 2w

\sf 60= 4w + 2w

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Find length:

\sf length = 2w

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5 0
2 years ago
Read 2 more answers
Help me plz <br><br>76° A B D с What is the measure of ZC? (Hint: just give numerical answer... ​
butalik [34]

Answer:

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Step-by-step explanation:

∠ACB is an inscribed angle while arcAB is its intercepted arc

An inscribed angle is equal to half the length of its intercepted arc

Hence, ∠ACB = 1/2 the measure of arc AB

arc AB = 76°

Hence ∠ACB = 1/2 of 76

76/2 = 38

Therefore ∠ACB = 38°

8 0
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marishachu [46]

Answer: 1/3divided by 3=x

Step-by-step explanation:

there is 1/3k of trail mix to divide among 3 people

hope this helped

plz mark brainliest so

i can rank up

;p <3

8 0
2 years ago
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