Average length of segments

[sallycats]

These unit semicircles contain n vertical segments drawn from the x-axis to the curve. In the
first diagram, the segments are drawn so that the diameter is partitioned into equal subintervals. In the
second diagram, the segments are drawn so that the semicircle is partitioned into equal subarcs. Let n
increase to infinity.

Should the average length of the segments in the first diagram to be greater than,
less than, or equal to the average length of the segments in the second diagram?

Find the average length of the segments in the first and second diagrams.

This feels like an integration problem, but I'm not sure where to start with this. Any hints or help would be appreciated!

 

[tutor_joel]

The second diagram segment lengths can be found using trigonometry. Each length in the second diagram creates the leg of a triangle with a hypotenuse of one. Simply figure out the angles which are evenly spaced through 180 degrees (15 degree increments) and you can find the segment lengths using the sine function.

thinking about the first one...

 

[tutor_joel]

Similarly, with the first diagram, you cannot find the angles, but you can find the horizontal leg of the triangles formed by the segments. Of course, the hypotenuses are always one. So you can use the pythagorean theorem to find the segment lengths in figure 1, knowing the hypotenuse and horizontal leg of each triangle.

The horizontal legs of each triangle will be 1/6, 2/6, 3/6 etc...

Let me know if this helps

 

[Deleted member 4993]

In the first case:

\(\displaystyle height (y) = \sqrt{1-x^2}\)

where x = 1/n

In the second case:

\(\displaystyle height = sin(\theta)\)

where:

\(\displaystyle \theta = \frac{360^o}{n}\)