How to find the average value of the function on the interval [1,8] ?

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Answer 1 · 2016-03-08T22:11:18

#"average value"=45/28#

The average value of the function #f(x)# on the interval #[a,b]# is

#"average value"=1/(b-a)int_a^bf(x)#

So, we know that

#"average value"=1/(8-1)int_1^8root3x#

Simplify the function so we can integrate it easily:

#"average value"=1/7int_1^8x^("1/3")#

To find the indefinite integral of #x^("1/3")#, use the rule:

#intx^n=x^(n+1)/(n+1)#

Yielding:

#"average value"=1/7[x^("4/3")/(4/3)]_1^8=1/7[3/4x^("4/3")]_1^8#

Evaluate:

#"average value"=1/7(3/4(8^("4/3"))-3/4(1^("4/3")))#

#"average value"=1/7(3/4(16)-3/4(1))#

#"average value"=1/7(45/4)#

#"average value"=45/28#