Find the average value of the function f(x)=x^3

Question

Find the average value of the function f(x)=x^3 over the interval [0,2] and find the value(s) of x at which the function assumes the average values 

Answer 1

 

The average value of f is defined as: 1/(b-a)∫ f(x) dx (where integral is evaluated from a to b)

 

 

If we are to integrate f(x) = x³  we get: (1/4)* (x⁴) 

 

 

Applying formula for average value: 

 

[1/(b-a)]*[(1/4)*(x⁴)]a to b

 

 

Evaluating this result where a = 0 and b = 2:

 

[1/(2-0)]*[(1/4)*(x⁴)]a to b

 

 

=(1/2)*[((1/4)*x⁴)]a to b

 

=(1/2)*[((1/4)*(2)⁴) - (2*(0^4))]

 

=(1/2)*[((1/4)*16)-0]

 

=(1/2)*(4)

 

=2

 

 

 

Answer 2

As per definition

Average value= [1/(2-0)]integral (x^3)dx.       take limits from 0 to 2

=(1/2)*(x^4)/4.        take limits from 0 to 2

=(1/2)*16/4

=4/2

=2