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Calculus/Volume problem
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Question
Hey Alon,
The following is a question I need help on. I would be great it if you could solve it in detail and explain it.
A sort of raindrop is obtained by revolving the profile curve
y = sqrt(x)*(x-C)^2 for 0<x<C
about the x-axis. Here C is a positive constant.
(a) Sketch the profile curve and the solid of revolution.
(b) For which value of C will the raindrop have volume 1? What are the approximate dimensions (length and diameter) of this raindrop?
Not sure you'll show me the answer to A. If you can, great. If you can't, no problem.
Thanks a lot for looking at this Alon.
Rob
The volume of revolution along the x-axis for a curve y=(x) is
)
defined to be V=π∫y²(x)dx. Where x goes from 0 to C.
Now we demand that the volume should equal 1. Thus,
1=π∫y²(x)dx=π∫[(√x)*(x-C)²]² dx=π∫x*(x-C)^4 dx
1=π∫x*(x-C)²*(x-C)² dx= π∫x*(x²-2xC+C²)(x²-2xC+C²)dx
1=π∫(x³-2x²C+xC²)(x²-2xC+C²)dx
1/π=∫[x^5-2x^4C+x³C²-2x^4C+4x³C²-2x²C³+x³C²-2x²C³+xC^4]dx
1/π=(1/6)x^6-(2/5)x^5C+(1/4)x^4C²-(2/5)x^5C+x^4C²-(2/3)x³C³+
+(1/4)x^4C²-(2/3)x³C³+(1/2)x²C^4 .
1/π=(1/6)x^6-(4/5)x^5C+(3/2)x^4C²-(4/3)x³C³+(1/2)x²C^4. Applying
the lower upper limits yields :
1/π=(1/6)C^6-(4/5)C^6+(3/2)c^6-(4/3)C^6+(1/2)C^6
(1/π)=(1/30)C^6 -> C=(30/π)^(1/6).
Alon.
P.S I attached you a sketch of the function.
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Alon Mandes
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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .
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1. I'm a team member of mathnerds (math site for answering questions)
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3. 2 years of experience as a math teacher in college.
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Mathematics & Physics.
5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" ,
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