What is wrong with this?

This question is usually posed as the length of the diagonal of a unit square. You start going from one corner to the opposite one following the perimeter and observe the length is 2, then take shorter and shorter stair-steps and the length is 2 but your path approaches the diagonal. So2√=2
In both cases, you are approaching the area but not the path length.
You can make this more rigorous by breaking into increments and
following the proof of the Riemann sum. The difference in area between
the two curves goes nicely to zero, but the difference in arc length
stays constant.

**ANSWER:-**This question is usually posed as the length of the diagonal of a unit square. You start going from one corner to the opposite one following the perimeter and observe the length is 2, then take shorter and shorter stair-steps and the length is 2 but your path approaches the diagonal. So

Edit: making the square more explicit. Imagine dividing the
diagonal into n segments and a stairstep approximation. Each triangle
is (1n,1n,2√n) . So the area between the stairsteps and the diagonal is n1n2 which converges to 0. The path length is n2n , which converges even more nicely to 2.

## 0 comments:

## Post a Comment

Don't Forget to comment