Question

What is the average distance between a point in the interval [-4,3] and the origin?

Answer 1

I get `25/14`

Explanation:

The distance between a point `x` on the number line and the origin is `absx`

We have been asked for the average value of this function on the interval `[-4,3]`

We can either evaluate `1/(3-(-4)) int_-4^3 absx dx` or work through the equivalent geometry without the integral.

The area under graph of `absx` and above `[=4,3]` consists of two isosceles right triangles.
On the left we have a triangle with base and height `4`, so its area is `8`.
On the right we have a triangle with base and height `3`, so its area is `9/2`.

The total area under the graph is `25/2`

Divide by the total base of `3+4 = 7` to get an average value of `25/14`