techniques of integration

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lvyral93

Mathematics

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the voltage v (in volts) induced in a tape head is given by v=t^2e^3t, where t is the time (in seconds). find the average value of v over the interval from t=0 to t=2. Round to the nearest volt.

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Explanation & Answer

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Given the voltage v=t^2 e^3t

to find the average value of v in the interval from t=0 to t=2 is

average value of v = integral  v  dt

                               = integral ( t=0 to t=2) t^2 e^3t  dt

                               = ( t=0 to t=2) (1 / 3)t^2*e^(3t) - (2 / 9)t*e^(3t) + (2 / 27)e^(3t)

     now substitute value of t


we get = [(1/3)(2)^2 * e^6 - (2/9)2  * e^(3 *2) + (2/27)e^(3*2) ] -[ (1/3)(0)^2 *e^(3*0) -(2/9)0*e^(3*0) +(2/27)e^(3*0)

          = (1/3)4*e^6 - (2/9)2e^6 + (2/27) e^6  -  [0 -0+(2/27)*1 ]

          =  4/3 * e^6 -4/9 e^6 +2/27e^6 -2/27

         = [ 4*9 e^6 - 4*3 e^6 +2 e^6 -2 ] / 27

        = [ 36 e^6 - 12e^6 + 2 e^6 - 2 ] / 27

       = [ 46 e^6 - 2]/27

so average value is [ 46 e^6 - 2]/27

Please let me know if you need any clarification. I'm always happy to answer your questions.


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