Description
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
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