sebadillab0

2022-07-12

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

treccinair

Beginner2022-07-13Added 18 answers

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

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

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

The area under graph of |x| 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 $\frac{9}{2}$

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 $\frac{25}{14}$

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

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

The area under graph of |x| 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 $\frac{9}{2}$

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 $\frac{25}{14}$

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