Find the average value of the function

Question

Find the average value of the function h ( r )= 15/(1+r)² on the interval [1,6].

h_ave=?

 

Answer 1

We must first find the integral of the function h.

 

h_ave  = ∫15 / (1 + r)² dr

 

We can rewrite this as

 

15∫(1 + r)^-2 dr

 

 

Lets apply the u-substitution method which uses the chain rule.

 

u = 1 + r

du = dr

 

Substituting these new variables into the integral, we get

 

15∫u^-2 du

 

= 15(-u^-1)

 

= -15/u

 

= -15/(1 + r)

 

Substituting the values 1 and 6 into this integral, we have

 

h_ave  = [-15/(1 + 6)] - [-15/(1 + 1)]

 

h_ave  = (-15/7) - (-15/2)

 

h_ave  = -15/7  +  15/2

 

h_ave  = (-30 + 105)/14

 

h_ave  = 75 / 14

 

h_ave  = 5.36