ijlal
  • ijlal
http://www.xtremepapers.com/CIE/International%20A%20And%20AS%20Level/9709%20-%20Mathematics/9709_s10_qp_13.pdf help me in Question no.9 pls reply quick thank you :)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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TuringTest
  • TuringTest
First we need the points A and B, know how to find that?
ijlal
  • ijlal
yes i know that
ijlal
  • ijlal
A has (1,5) and B(4,5)

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ijlal
  • ijlal
M(2,4)
TuringTest
  • TuringTest
right, so our integration will be from 1 to 4 what is the outer radius of each disk we will use in our method? what is the radius of the inner disk?
ijlal
  • ijlal
i dont know how a cylinder is formed in this curve
TuringTest
  • TuringTest
we are making rings out of each section of area|dw:1327072712405:dw|the outer radius of each ring will be\[r_0=5\]the inner radius will be\[r_i=x+\frac{4}{x}\]the area of each disk wile therefor be\[\pi r_0^2-\pi r_i^2\to\pi(r_0^2-r_i^2)\]
TuringTest
  • TuringTest
|dw:1327073020023:dw|here is a cross-section of each ring|dw:1327073083149:dw|therefor adding up these rings we will integrate the areas of the rings from x=1 to x=4\[\pi\int_{1}^{4}r_0^2+r_i^2dx=\pi\int_{1}^{4}5^2-(x+\frac{4}{x})^2dx\]make sense?
TuringTest
  • TuringTest
*slight typo above, should be minus in the first integral (because we are subtracting the area of the inner disk from that of the outer)\[\pi\int_{1}^{4}r_0^2-r_i^2dx=\pi\int_{1}^{4}5^2-(x+\frac{4}{x})^2dx\]
ijlal
  • ijlal
Yes the Volume of the ring is i guess 57pie
TuringTest
  • TuringTest
I guess, I haven't checked. I'll leave the integration for you to do. But your final answer is a volume of a shape, each section is a ring. (Just pointing out the discrepancy in calling the whole thing a ring)
ijlal
  • ijlal
the answer of the question is 18pie i dont know how the cylinders height is 4units can you tell me that i will be done with my question done
TuringTest
  • TuringTest
which cylinder has a height of 4 units?
ijlal
  • ijlal
there is only one cylinder formed
TuringTest
  • TuringTest
\[\pi\int_{1}^{4}r_0^2+r_i^2dx=\pi\int_{1}^{4}5^2-(x+\frac{4}{x})^2dx=\pi\int_{1}^{4}25-x^2-8-\frac{16}{x^2}dx\]\[=\pi(17x-\frac{x^3}{3}+\frac{16}{x})|_{1}^{4}\]\[=\pi[(68-\frac{16}{3}+4)-(17-\frac{1}{3}+16)]\]whatever that simplifies to... The whole shape will have a length of 3 units (from x=1 to x=4) if that is what you mean....
TuringTest
  • TuringTest
I got\[\pi\int_{1}^{4}r_0^2-r_i^2dx=\pi\int_{1}^{4}5^2-(x+\frac{4}{x})^2dx=\pi\int_{1}^{4}25-x^2-8-\frac{16}{x^2}dx\]\[=\pi(17x-\frac{x^3}{3}+\frac{16}{x})|_{1}^{4}\]\[=\pi[(68-\frac{16}{3}+4)-(17-\frac{1}{3}+16)]=32\pi\]and you say that you know the answer to be 18pi ?
ijlal
  • ijlal
yes the answer is 18 pi it is given in the mark scheme of this question
ijlal
  • ijlal
and this solution simplifies to 34pi not 32pi
TuringTest
  • TuringTest
Oh, I just made some silly arithmetic mistake apparently (this is why I left the integral for you to do) As you can see by wolfram it is set up right\[\pi\int_{1}^{4}r_0^2-r_i^2dx=\pi\int_{1}^{4}5^2-(x+\frac{4}{x})^2dx\]http://www.wolframalpha.com/input/?i=int%20from%201%20to%204%20(25-(x%2B4%2Fx)%5E2)dx&t=crmtb01 so you figure out where I messed up...
ijlal
  • ijlal
the solution of this integral gives 90pi
TuringTest
  • TuringTest
no, look at the link I posted. Wolfram does no lie: http://www.wolframalpha.com/input/?i=pi*int+from+1+to+4+%285%5E2-%28x%2B4%2Fx%29%5E2%29dx there it is explicitly, clearly the answer to the integral I wrote is 18pi
ijlal
  • ijlal
oh sorry i neglected the minus sign by mistake :P thank you :)
TuringTest
  • TuringTest
very welcome :) I apparently made a similar mistake trying to do it by eye, so it happens.

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