Question

Find the average value of the function y = 6 - x2 over the interval [-1, 4]

Answer 1

The average value of the function
`f(x) = 6 - x^(2)` over the interval
`[-1,4]` is
`(5)/(3)`.

Explanation:

The average value of a function over an interval is represented by this integral:

`\bar y = (1)/(b-a)\cdot \int\limits^(b)_(a) {f(x)} \, dx`

Where:

`a`,
`b` - Lower and upper bounds of the interval, dimensionless.

`f(x)` - Function, dimensionless.

If
`a = -1`,
`b = 4` and
`f(x) = 6 - x^(2)`, the average value of the function is:

`\bar y = (1)/(4-(-1))\int\limits^(4)_(-1) {6-x^(2)} \, dx`

`\bar y = (6)/(5)\int\limits^(4)_(-1) \, dx - (1)/(5)\int\limits^(4)_(-1) {x^(2)} \, dx`

`\bar y = (6)/(5)\cdot x |_(-1)^(4) - (1)/(15)\cdot x^(3)|_(-1)^(4)`

`\bar y = (6)/(5)\cdot [4-(-1)]- (1)/(15)\cdot [4^(3)-(-1)^(3)]`

`\bar y = (5)/(3)`

The average value of the function
`f(x) = 6 - x^(2)` over the interval
`[-1,4]` is
`(5)/(3)`.