Question

How do you use the integral fomula to find the average value of the function `f(x)=18x` over the interval between 0 and 4?

Answer 1

I found: `y_(av)=36`

Explanation:

Consider a general case:

The average value of your function `y_(av)` will be the one that makes the area 1 exactly equal to area 2: So the area under the curve will be exactly the area of the rectangular region (blue+area 1).
The rectangular area `A` will be:
`A=basexxheight=(x_2-x_1)*y_(av)=int_(x_1)^(x_2)f(x)dx`
and so:
`color(red)(y_(av)=1/(x_2-x_1)*int_(x_1)^(x_2)f(x)dx)`

In your case:
`y_(av)=1/(4-0)*int_(0)^(4)18xdx=18/4[x^2/2]_0^4`

so finally:

`y_(av)=36`

Graphically: