Average Value

Question

Find the average value of  f(x) =1/x over the interval [1, 3].
  
0.67

1.10

0.55

0.33

Answer 1

By definition of f(x) = ln x, we have ∫₁³ dx/x = ln 3.

 

So the average is that divided by 3-1=2: ln√3 ≈ 0.55

Answer 2

At x=1, y =  f(x) = 1/1 = 1     the point is (1, 1)

 

At x=3, y = f(3) = 1/3          the point is (3, 1/3)

 

The average value of y [that is, of f(x)] over this interval is:

      (1 + 1/3) / 2

           (4/3) / 2

           2/3      (or, 0.67)