FIND AVERAGE VALUE OF PIECEWISE FUNCTION OVER AN INTERVAL

Let f(x) be a function on the interval [a,b]. If we divided our interval into n equally sized intervals and took the sample at left endpoints f(xi), then an approximation for the average value would be given by the formula:

average-value-of-function

Find the average value of the function over the given interval

Problem 1 :

average-value-of-piecewise-funq1.png

Solution :

average-value-of-piecewise-funq1p1.png

Finding average value of left piece :

f(x) = -x2 - 6x - 8 on [-4, -3]

a = -4, b = -3

fave = 1(b-a)baf(x) dx= 1(-3-(-4))-3-4-x2-6x-8 dx= 1(-3+4)-3-4-x2-6x-8 dx=-x2+12+1-6x1+11+1-8x-3-4=-x33-6x22-8x-3-4= -(-3)33-6(-3)22-8(-3)--(-4)33-6(-4)22-8(-4) = 273-6(9)2+24--(-64)3-6(16)2+32= (9+24-27)-(21.3-48+32)= 6-5.3= 0.7

Finding average value of right piece :

f(x) = (-x/2) - (1/2) on [-3, 3]

a = -3, b = 3

fave = 1(b-a)baf(x) dx= 1(3-(-3))3-3-x2-12 dx=16-x2+12(2+1)-12x3-3=16-x36-12x3-3= 16-(3)36-12(3)- -(-3)36-12(-3)= 16-276-32- 276+32= 16-92-32- 92+32= 16(-6-6)= -2

Adding these two values,

= 0.7 + (-2)

= -0.3

So, the average value of the given function in the given interval is -0.3.

Problem 2 :

average-value-of-piecewise-funq2.png

Solution :

average-value-of-piecewise-funq2p1.png

Finding average value of left piece :

f(x) = -1 on [0, 1]

a = 0, b = 1

fave = 1(b-a)baf(x) dx= 1(1-0)10(-1) dx=(-x)10

Finding average value of right piece :

f(x) = -x2 + 4x -4 on [1, 3]

a = 1, b = 3

fave = 1(b-a)baf(x) dx= 1(3-1)31-x2+4x-4 dx=12-x2+1(2+1)+4x1+11+1-4x31=12-x33+4x22-4x31=12-333+4(3)22-4(3)--133+4(1)22-4(1)=12((-9+18-12)-(=12(-3-(

= -1 + (-0.35)

= -1.35

So, the average value is -1.35.

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