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

1 Answer
Jim H
Jun 7, 2017

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#

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