The average value of f(x)

Question

If the average value of f(x) = x² + 6kx over the interval [0, 6] is 84, then k = 

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4.

24.

Answer 1

The average value of a function in some domain is the definite integral of that functon over the domain, divided by the size of the domain. In other words

<f(x)>=1/(x_f-x_i) ∫_xi ^xf f(x) dx

For our purposes, we have x_i=0, x_f=6, and f(x)=x²+6kx. So, we can charge forward

<f(x)>=1/(6-0)∫₀ ⁶ x²+6kx dx=1/6[x³/3+3kx²]|₀ ⁶=1/6[6³/3+3k6²]=6[2+3k]

We are told this average value is 84 for this function, so we can use that to solve for k

6[2+3k]=84

2+3k=84/6=14

3k=12

k=4