Question

What is the average value of the function `y= 3x^2 - 2x` on the interval `[2,4]`?

Answer 1

Average value `= 22`

Explanation:

The average value of `y=3x^2-2x` over the interval `[2,4]`
is the area under the curve divided by the width of the interval.

The area under the curve for the interval `(2,4)` is
`color(white)("XXX")A=int_2^4(3x^2-2x) dx`

`color(white)("XXXx")=int_2^4(3x^2)dx - int_2^4(2x)dx`

`color(white)("XXXx")=x^3]_2^4 - x^2]_2^4`

`color(white)("XXXx")=(64-8) - (16-4)`

`color(white)("XXXx")=56-12 = 44`

The width of the interval `[2,4]` is `4-2=2`

So the average value of `y` over this interval is `44/2=22`