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

Jun 7, 2017

I get $\frac{25}{14}$

#### Explanation:

The distance between a point $x$ on the number line and the origin is $\left\mid x \right\mid$

We have been asked for the average value of this function on the interval $\left[- 4 , 3\right]$

We can either evaluate $\frac{1}{3 - \left(- 4\right)} {\int}_{-} {4}^{3} \left\mid x \right\mid \mathrm{dx}$ or work through the equivalent geometry without the integral.

The area under graph of $\left\mid x \right\mid$ and above $\left[= 4 , 3\right]$ 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 $\frac{9}{2}$.

The total area under the graph is $\frac{25}{2}$

Divide by the total base of $3 + 4 = 7$ to get an average value of $\frac{25}{14}$