Mathematics 1

x2 x3,

Hence: Area A=dx=

=(-)=--0=

In the Area B )

x3 x2,

Hence: Area B=dx=()=-

-+==

The total area is: +==

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Average value of a Function

• The average (mean) value of a function over the interval
• a≤x≤b is given by :
• faverage(x)=
• Assume f(x)=
• To check find the average value by 2 ways:
• Average value:==2
• Average value = =++(2+4+6)=2 – the same.

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Average value of a Function -proof

• The average value of n numbers is the sum of all the numbers divided by n . Let’s take the interval [a,b] and divide it into n subintervals each of length ∆x= , then n= (1)
• Now from each of these intervals choose the points
• We can then compute the average of the function values
• by computing, ……… (2)

;

• Plugging n from (1) into (2) we get:
• Increasing n and taking limit we get:
• the limit of the sum is the definition of the definite integral.
• Q.E.D.

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Average value of a Function - Ex1.53

• Rectification transforms
• An AC voltage V (t) = V0 sin(ωt) (with amplitude V0 > 0 and angular frequency ω) into the all positive rectified voltage VR(t) = V0 | sin(ω t)|. This is then fed into a smoothing circuit to produce a DC output which is the average of the rectified voltage over complete cycles. Find the DC output voltage VDC.
• Solution:

DC voltage - average of VR(t) over

1 cycle

• ω t= the cycle for the

rectified voltage t=- cycle

Length (period)

VDC=

==(-cos= (cos- cos=(1-(- 1))=2≈0.6366 V0

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AC voltage
0
V0
-V0
V0
0
VR(t) voltage

Average value of a Function –more examples

• Ex1.54
• A 2m long metal rod is heated resulting in a temperature T(x) ◦C at a position x metres from the end where
• T(x) = What is the average temperature?
• Solution:
• Taverage==-(1+x)-1=-(-1)==25 ◦C
• Ex1.55
• The amount of radioactive material after t hours in a sample is given by R(t) =105e(−t/10). Find the average value of R(t) over (i) the first 24 hours and (ii) the first and last hour of this period. Work to three significant figures.
• Solution:
• (i) R Average over 24 hours:=dt= (-10 ) =(+1)= 3.79
• (ii) R Average over the last hour=dt=(-10 ) =(+ )= 9540=9.540

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