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Expert: Alon Mandes - 10/1/2009


Question
I need to find the volume obtained by rotating the region bounded by the curves y=sqrt(x) and y=x.

I think it makes a donut shape cross section when you draw the graph out and rotate it.  But I am unclear what to do from here.  I know I need to find the two radii. But thats as far as I can go really.


Answer
1st of all, let's find the point of intercept of both curves :
√x=x --> x=0 & x=1 . That means that the limits of the integral will be from 0 to 1 .
2nd, the volume obtained by rotating is defined as :
  b
V=π∫[f(x)-g(x)]² dx . Therefore :
a
  1
V=π∫[x-√x]² dx =
 0
  1
V=π∫[x²+x-2x√x] dx =
 0
  1
V=π∫[x²+x-2x^(3/2)] dx =
 0
V=π * (1/3)x^3 + (1/2)x^2 - (4/5)x^(5/2) {form 0 --> 1}
V=π * (1/3)+(1/2)-(4/5)= 0.033π

Alon.  

Calculus

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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) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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M.A in Mathematics & Bs.c in Electronics.

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