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Integral Calculus

abhishek sinha's Avatar
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11 May 2008 13:04:02 IST
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Challenge in Integration - From Feynmann
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Solve the following in the shortest possible way



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abhishek sinha's Avatar

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12 May 2008 19:26:43 IST
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a hint !!




 


this problem  WOULD NOT REQUIRE ANY TYPE OF SUBSTITUTION OR BY PARTS ETC .




 


THE ULTIMATE INTEGRATION IS JUST A FORMULA  AVAILABLE IN THE FIRST PAGE OF ANY INTEGRATION BOOK .




 


IT JUST REQUIRES  A LI'L BIT OF IMAGINATION !!

pink_ele's Avatar

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Joined: 16 Jan 2007
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14 May 2008 12:28:03 IST
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here a>b>0



let a side X forms triangle wid a and b den







dis gives



a^2+b^2-2abcosx=X^2



ryt



now



(sinx/X)^2=k(constant)................constant=(1/4R^2)



thus given integral is integral of a constant



integral=pi x constant



thats it !!!!!!1



rest ........still m solving

abhishek sinha's Avatar

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14 May 2008 18:10:01 IST
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You are very close !!


Please complete the remaining steps !!

pink_ele's Avatar

Blazing goIITian

Joined: 16 Jan 2007
Posts: 1836
14 May 2008 18:20:49 IST
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bhaiyya
u mean to prove R=a/rt 2
fyn......
i'll try:)
abhishek sinha's Avatar

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15 May 2008 12:27:02 IST
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Now I am providing the complete solution to this problem :


For ease of understansing , let us change the variable x to C


Now consider a triangle whose sides are a, b ( fixed )(a>b) and their intermediate angle C is variable .


Now it is clear that the Dr. of the given integrand =



so the integral  now becomes



Now applying Sine rule , we know that



again we have that       dC = - dA -dB


Now let us imagine what would be the limits of A & B as C varies from 0 to ( with a>b ) .


when C->0  , A-> & B->0


again when C->, A->0 & B->0 { Draw a figure with a>b & observe the above result yourself !! }


so the integral becomes now



The last integral is clearly zero !!


So the ultimate integral is



but a is constant ; so the result is


=


 


( hence proved )




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